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ItOBLN SOM’S M A TII £ M A TI CAL 


SERIES. 


A 


NEW TREATISE 

ON 



THEORETICAL AND PRACTICAL: 


WITH 

* 

I 

tJSE OF INSTRUMENTS, ESSENTIAL ELEMENTS OF TRIGONOMETRY, 

AND THE NECESSARY TABLES, 


FOR 

SCHOOLS. COLLEGES, AND PRACTICAL SURVEYORS. 

or A' o neXsoirv • A\oVo\'r\ £ ov\ 

EDITED I> V 

OREN ROOT, 

* ' 

PROFESSOR OF MATHEMATICS IN HAMILTON COI.LKO 

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APR 9 1891 

2 7 SD 

A 1 /T 3H WG 

A . M., 


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NEW YORK CINCINNATI CHICAGO 

AMERICAN BOOK COMPANY 


FROM THE PRESS OF 
IVISON, I.U.AKEKAN A" C( 51TANV 



Copyright, 1863 and 1891, by D. W. Fish. 



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<o-?S 7 



EDITOR’S PREFACE. 


In the preparation of the present edition of Surveying and 
Navigation, no effort has been made to preserve the identity 
of the former work; but the aim has been to prepare an 
entirely new treatise , far more complete than the former, not 
only in the range and variety of topics, but also in improved 
methods and practical applications, and to combine the best 
'practical with the highest theoretical character, making the 
work worthy a place in the Series for which it was intended. 

The object kept in view by the editor has been two-fold. 

To prepare a work suitable for class instruction in schools 
and colleges, furnishing clear rules with plain explanations, 
and an abundance of examples and illustrations. 

2d. To prepare a work valuable as a book of reference for 
the practical surveyor, containing all necessary tables, and all 
processes required for any practical operations. 

The chapter on Trigonometry is taken mainly from Robin¬ 
son’s New Geometry and Trigonometry, as the subject was 
there treated with such clearness that few changes seemed 
desirable. 

The more practical rules of Mensuration, even though com¬ 
mon, have been given, with examples, in order to make the 
sections as complete as possible, and to make the book useful 
to young students as well as to those more advanced. 

The Examples in the Chapters on Land Surveying have been 
formed mainly from the field notes of actual operations, and 
are sufficiently numerous to familiarize any student with the 
working of all practical cases. 

The section upon Division of Land is also made up to a 
great extent from notes of actual surveys, and gives ample 
illustration of methods, to enable any one with a good knowl¬ 
edge of mathematical principles to master any case that 
mav come before him. In the Division of Land, cases are so 


IV 


PREFACE. 


diverse, that mathematical principles, and not specific rules 
must be relied upon, mainly, by the surveyor. 

The subjects of Leveling, Road and Canal Surveying, and 
Topography, belong more properly to civil engineering, and a 
full discussion of these topics would require separate volumes. 
The sections upon these subjects are intended, therefore, to 
furnish only the more elemental and simple points. 

The chapter on Navigation is intended for class instruction 
merely, as extensive works, with more complete tables, are 
necessary for the practical navigator, such as are supplied by 
Bowditch and others. A method of obtaining difference of 
longitude, not given in any of the text-books, and also a 
method of obtaining meridional parts have been given. 

It has been the intention of the editor so to arrange the 
work that a student, who did not desire a full course of trigon¬ 
ometry, could, after the study of geometry by the use of this 
text-book, obtain the necessary propositions for the practical 
surveyor in clear and concise form, and thus fit himself more 
speeddy T for practical work. 

The arrangement of the work, including as it does Trigo¬ 
nometry and Mensuration, requires that two terms should be 
employed in its completion; those students, however, who 
have before studied Trigonometry, by omitting Chapter II. 
and also Section III. of Chapter III. can readily master 
the Surveying proper in one term. 

It is just to acknowledge here the services, as co-editor in 
the preparation of this work, of Oren Root jr., A.M., of Rome, 
N.Y., whose attainments as a mathematical scholar, and 
experience as a teacher are well-known. 

This treatise is now submitted to the public, with the hope 
that it will fully commend itself, for fulness of matter, and 
scientific arrangement; for clear statement, and accurate defi¬ 
nition • for rules concise and of easy application ; for exam- 
pies numerous, apt, and strictly practical ; for whatever, in 
short, the student, and the practical surveyor could reasonably 
expect in a first class text-book on this subject. 

Hamilton College, Sept. 1st, 1863. 


V 



TABLE OF CONTENTS. 


CHAPTER I. 
INSTRUMENTS. 
SECTION I. 


PAQB 

Surveying Instruments. 9 

The Surveyor’s Compass. 10 

The Solar Compass . 13 

The Transit. 24 

The Surveyor’s Chain.. 27 

The Plane Table. 27 

The Engineer’s Level. 31 

The Leveling Rod. 33 

The Theodolite. 34 


SECTION II. 


Instruments in Navigation,. 

The Sextant. 35 

The Log.. 41 

SECTION III. 

Plotting Instruments. 42 

The Diagonal Scale of Equal Parts. 44 

Gunter’s Scale. 44 

The Sector. 46 


SECTION IV. 


Problems solved Instrumentally 


48 




















vi 


TABLE OF CONTENTS. 


CHAPTER II. 

LOGARITHMS AND TRIGONOMETRY. 

SECTION I. 

TAGE 

Logarithms. 62 

Logarithmetic Table. 65 

SECTION II. 

Plane Trigonometry. 76 

Definitions. 77 

Tables of Sines, Co-sines, etc. 93 

Natural Sines, etc. 93 

Logarithmic Sines, etc. 94 

Practical Applications. 96 

Practical Problems. 102 

Oblique-angled Plane Triangles.. 103 

Practical Problems. 112 

* SECTION III. 

Spherical Trigonometry. 114 

Right-angled Spherical Trigonometry. 115 

Napier’s Circular Parts. 119 

Solution of Right-angled Spherical Triangles. 121 

Practical Problems. 124 

Quadrantal Triangles. 126 

Practical Problems. 129 

Oblique-angled Spherical Trigonometry. 130 

Solution of Oblique-angled Spherical Triangles. 142 

Practical Problems. 147 

CHAPTER III. 
MENSURATION. 

SECTION I. 

Mensuration of Lines. 151 

Of Heights and Distances... 152 

Practical Problems. 173 


























TABLE OF CONTENTS. vii 

SECTION II. 

PAGH 

Mensuration of Surfaces. 177 

SECTION III. 

' t 

Mensuration of Solids. 198 

0 

CHAPTER IV. 

LAND SURVEYING. 

SECTION I. 

Definitions... 220 

Traverse Table. 223 

SECTION II. 

Measurement of Lines and Angles. 224 

SECTION III. 

Measurement of Areas. 226 

SECTION IV. 

Rectangular Surveying. 238 

SECTION V. 

Dividing and Laying out Land. 258 

SECTION VI. 

Triangular Surveying. 283 

Harbor Surveying. 287 

SECTION VII. 

Canal and Road Surveying. 289 

Public Lands. 290 

Survey Bill. 291 

SECTION VIII. 


Variation of the Compass 
Table of Declinations.... 


292 

299 

















viii 


TABLE OF CONTENTS,! 


CHAPTER Y. 

'TOPOGRAPHICAL SURYEYIHG. 

SECTION I. 

PAGE 

Locating Curves. 303 

SECTION II. 

Leveling. 306 

Contour of Ground.... 314 

Elevations determined by the Barometer. 316 

Practical Applications. 320 

CHAPTER VI. - . 

- *■ j * * + * •? - ■ -■*», . . . . ... . + 

NAVIGATION. 

SECTION I. 

Definitions.. 325 

Leeway. 327 

Card of the Mariner’s Compass. 328 

SECTION II. 

P.ane Sailing. 329 

SECTION III. 

To find the Difference of Longitude. 337 

SECTION IY. 

Mercator’s Sailing. 349 

Direct Method of Computing Meridional Parts. 355 

SECTION Y. 

Sailing in Currents. 35 

Parallax. 30i. 

4 

SECTION VI. 

Latitude. 363 

SECTION VII. 

Longitude. 370 

SECTION VIII. 

Lunar Observations. 376 





















SURVEYING AND NAVIGATION 


CHAPTER I. 

INSTRUMENTS. 

SECTION I. 

♦ 

SURVEYING INSTRUMENTS. 

In Mensuration, Surveying, and Navigation, the known 
quantities or conditions are determined by measurements. 

The more common and important instruments used for these 
measurements are the Surveyor’s Compass, the Solar Compass, 
the Transit, the Level, the Plane Table, the Sextant, and vari¬ 
ous Plotting Instruments. These are described in the present 
chapter, and rules or directions given for their use. 

It is hoped that the descriptions given will prove to be 
clear, and readily understood. Yet both teacher and student 
should remember that it is impossible to gain a perfect idea 
of a complex instrument by a description merely. In all pos¬ 
sible cases, the instrument itself should be studied in connec¬ 
tion with the description; the various adjustments should be 
made in presence of the student, and some practice given in 
the use of the instruments. Surveying and Navigation are 
important applications of geometrical science, and care should 
be taken always to secure to the student, as far as possible, the 
practical objects of a study of these branches. 



10 


SURVEYING AND NAVIGATION. 


SURVEYOR’S COMPASS. 



The Surveyor’s Compass consists essentially of a graduated 
circle, in the centre of which a magnetic needle is suspended, 
so as to turn freely in a horizontal direction. 

The compass plate has two spirit levels placed at right 
angles to each other to level the compass in both directions, 
and at its ends standards or sights through the slits of which 

o o 

the compass is directed to the object sighted upon. 

The graduated circle is divided into degrees and half de- 
grees, and figured from 0 to 90 on each side of the north and 
south ends of the compass box ; the line of the sights passes 
through the zero divisions of the circle. 

In the best compasses also the circle is made to turn about 
its centre, in order that the line of zeros may be moved a short 
distance to either side of the line of sight, that allowance may 
be made for the variation of the needle , and to read the needle 
to single minutes of a degree, as will be hereafter explained. 

In the compass shown in the cut, the movement of the circle 
is effected by a pinion at a , working into a circular rack on 
the outside of the compass box. The space over which the 
circle is moved is shown in degrees and minutes by a divided 
arc and vernier , near the letter S. 
























INSTRUMENTS. 


II 


The arc is divided to degrees and half-degrees, and figured 
each way from its centre. The vernier is divided into thirty 
equal spaces, which together exactly correspond in length with 
twenty-nine half-degrees of the arc. 

Each division of the vernier is therefore one-thirtieth, or i:i 
other words, one minute longer than a single division of tht: 
arc, and the vernier thus reads to single minutes. 

The compass shown in the cut has also a horizontal circle 
beneath the main plate, read by two opposite verniers, by 
which horizontal angles can be taken to single minutes with¬ 
out the use of the needle. 

Adjustments :— 

1. Bring the bubbles into the centre by the pressure of the 
hand on different parts of the plate, and turn the compass 
half way around; should the bubbles run to the ends of the 
tubes, it would indicate that those ends were the highest; 
lower them by tightening the screws immediately underneath, 
and loosening those under the lower ends, until by estimation, 
half the error is removed ; level the plate again, and repeat the 
operation until the bubbles will remain in the centre during 
an entire revolution of the compass. 

2. The sights are tested by a plumb line, with which their 
slits must coincide when the compass is horizontal. If they 
do not, file off the under side of the base, until the connection 
is made. 

3. The needle should cut opposite degrees in any part of 
the circle. If it does not, bend the centre pin until it will 
cut at a given degree, say 0 or 90. Then holding the needle 
in the same position, turn the compass half way around, and 
note whether it now cuts on precisely the same degrees at 
both ends ; if not, correct half the error by bending the needle, 
and the remainder by bending the centre pin. Repeat this 
operation until the needle is perfectly straight, when it may 
be brought into line with all other divisions of the circle by 
bending the centre pin. 


12 


SURVEYING AND NAVIGATION. 


1. To use the Surveyors Compass. 

Place the instrument on its staff or tripod, level the com¬ 
pass plate, lower the needle upon its pivot, and direct the 
sights to any object the bearing of which is desired, always 
keeping the south end nearest the person. 

Wait until the needle is perfectly at rest, and read the 
bearing from the north end of the needle. 

2. To turn off the Variation of the Needle. 

Place the compass on some well-defined line of the old sur¬ 
vey, and turn the pinion, a , until the needle indicates the 
same bearing as that given in the field notes of the original 
survey ; the reading of the vernier will give the change of 
variation during the period which has elapsed since the origi¬ 
nal survey. 

3. To read to Minutes by the Vernier. 

First make the zero of the vernier coincide with that of the 
divided arc; then noting the number of whole degrees given 
by the needle, move back the compass circle until the nearest 
whole degree mark is made to coincide with the point of the 
needle; count the minutes from the zero point of the vernier, 
until a division on the vernier is found exactly in line with 
another on the divided arc, and this reading, added to the 
whole degrees, will give the bearing to single minutes. 

Note. —The Surveyor’s Compass, the Solar Compass, the Surveyor’s 
Transit, and the Y Level, as represented and described in this chapter, are 
manufactured by the well-known firm of W. & L. E. Gurley, Troy, N. Y., 
whose instruments for accuracy, adaptation, and finish, are unsurpassed by 
any others made in this country. For further information, price, etc., see 
advertisement at the end of this work. 


\ 


V 


■ 

. 




























. 

. 













































13 


INS TRUMENTS, 


THE SOLAR 'COMPASS 


This instrument, used with the sun in determining meridians, 
or true north and south lines, is always employed in tracing 
the important lines of government lands. 

The Solar Apparatus consists mainly of three arcs of circles, 
by which can be set off the latitude of a place, the declination 
ot the sun, and the hour of the day. 

These arcs, designated in the cut by the letters, b , and c , 
are therefore termed the latitude, the declination, and the hour 
arcs respectively. 

T1 ie Latitude Arc, a , has its center of motion in two pivots, 
one of which is seen at d, the other is concealed in the cut. 

It is moved either up or down within a hollow arc, seen in 
the cut, by a tangent screw at f , and is securely fastened in 
any position by a clamp screw. 

The latitude arc is graduated to quarter degrees, and reads 
by its vernier, <?, to single minutes; it has a range of about 
thirty-five degrees, so as to be adjustable to the latitude of any 
place in the United States. 

The Declination Arc, b , is also graduated to quarter degrees, 
and has a range of about twenty-four degrees. 

Its vernier, -y, reading to single minutes, is fixed to a mov¬ 
able arm, A, having its center of motion in the center of the 
declination arc at g; the arm is moved over the surface of the 
declination arc, and its vernier set to any reading by turning 
the head of the tangent screw, A*. It is also securely clamped 
in any position by a screw, concealed in the engraving. 

Solar Lenses and Lines .—-At each end of the arm, A, is a 
rectangular block of brass, in which is set a small convex lens, 
having its focus on the surface of a little silver plate, fastened 
by screws to the inside of the opposite block. 

The silver plate, with its peculiar lines, will be referred to 
more particularly hereafter. 

The Hour Arc, c, is supported by the two pivots of tliedati-. 


/ 



14 


SURVEYING AND NAVIGATION. 


tude arc, already spoken of, and is also connected with that 
arc by a curved arm, as shown in the figure. 

The hour arc has a range of about 120°, is divided to half 
degrees, and figured in two series; designating both the hours 
and the degrees, the middle division being marked 12 and 90 
on either side of the graduated lines. 

The Polar Axis .—Through the center of the hour arc passes 
a hollow socket, p, containing the spindle of the declination 
arc, by means of which this arc can be moved from side to 
side over the surface of the hour arc, or turned completely 
round as may be required. 

The hour arc is read by the lower edge of the graduated side 
of the declination arc. 

The axis of the declination arc, or indeed the whole socket, 
p, is appropriately termed the polar axis. 

The parts just described constitute properly the solar a_p- 
paratus. 

Besides these, however, are seen the needle box, n , with its 
arc and tangent screw, t , and the spirit levels, for bringing the 
whole instrument to a horizontal position. 

The Needle Box, has an arc of about 36° in extent, divided 
to half degrees, and figured from the center or zero mark on 
either side. 

The needle, which is made as in other instruments, except 
that the arms are of unequal lengths, is raised or lowered by a 
lever shown in the cut. 

The needle box is attached by a projecting arm to a tangent 
screw, t , by which it is moved about its center, and its needle 
set to any variation. 

This variation is also read off by the vernier on the end of 
the projecting arm, reading to single minutes a graduated arc 
attached to the plate of the compass. 

Lines of Refraction .—The inside faces of the sights are also 
graduated and figured, to indicate the amount of refraction to 
be allowed when the sun is near the horizon. These are no* 
shown in the cut. 


INSTRUMENTS. 


15 


The Horizontal Limb in all Solar Compasses is divided upon 
silver, and reads by two opposite verniers to single minutes of 
a degree, the number of minutes being counted off in the same 
direction in which the vernier moves. 


PRINCIPLES OF THE SOLAR COMPASS. 


We are now prepared to proceed to the explanation of the 
2>eculiar construction of the instrument we are considering. 

The little silver plate before referred to, is shown in detail 
in the following figure. On the surface are marked two sets 

of lines intersecting each other at right 

Cj n 


0 


E3 






angles; of these, b b are termed the hour 
lines, and c c, the equatorial lines, as haw 
ing reference respectively to the hour of 
the day and the position of the sun, in relation to the equator. 

Below the equatorial lines are also marked three other lines, 
which are five minutes apart, and are of service in making 
allowance for refraction, as will be hereafter explained. 

The interval between the two lines, <?, c, as well as between 
A, b , is just sufficient to include the circular image of the sun, 
as formed by the solar lens, on the ojjposite end of the revolv¬ 
ing arm. 


When, therefore, the instrument is made perfectly horizon¬ 
tal, the equatorial lines and the opposite lenses being accurately 
adjusted to each other by a previous operation, and the sun’s 
image brought within the equatorial lines, his position in the 
heavens, with reference to the horizon, will be defined with 
precision. 

Suppose the observation to be made at the time of one of 
the equinoxes; the arm, A, set at zero on the declination arc, A, 
and the polar axis,j9, placed exactly parallel to the axis of the 
earth. 

Then the motion of the arm, A, if revolved on the spindle of 
the declination arc around the hour circle, c, will exactly cor¬ 
respond with the motion of the sun in the heavens, on the given 
day and at the place of observation ; so that if the sun’s image 










16 


SURVEYING AND NAVIGATION. 


was brought between the lines, c <?, in the morning, it would 
continue in the same position, passing neither above nor below 
the lines, as the arm was made to revolve in imitation of the 
motion of the sun about the earth. 

In the morning as the sun rises from the horizon, the arm, 
A, will be in a position nearly at right angles to that shown in 
the cut, the lens being turned towards the sun, and the silver 
plate on which his image is thrown directly opposite. 

As the sun ascends, the arm must be moved around, until 
when he has reached the meridian, the graduated side of the 
declination arc will indicate 12 on the hour circle, and the arm, 
A, the declination arc, &, and the latitude arc, a , will be in the 
same plane. 

As the sun declines from the meridian, the arm, A, must be 
moved in the same direction, until at sunset its position will 
be the exact reverse of that it occupied in the morning. 

Allowance for Declination. —Let us now suppose the ob¬ 
servation made when the sun has passed the equinoctial point, 
and when his position is affected by declination. 

b>- referring to the Almanac, and setting off* on the arc his 
declination for the given day and hour, we are still able to 
determine his position with the same certainty, as if he re¬ 
mained on the equator. 

When the sun’s declination is south, that is, from the 22d 
of September to the 20th of March in each year, the arc, £, is 
turned towards the plates of the compass, as shown in the 
engraving, and the solar lens, o , with the silver plate opposite, 
are made use of in the surveys. 

The remainder of the year, the arc is turned from the plates, 
and the other lens and plate employed. 

When the Solar Compass is accurately adjusted, and its 
plates made perfectly horizontal, the latitude of the place, 
and the declination of the sun for the given day and hour, 
being also set off on the respective arcs, if the image of the 
sun is brought between the equatorial lines the polar axis will 
be in the plane of the meridian of the place , or in a position 


INSTRUMENTS. 


IT 


parallel to the axis of the earth , and the sights will indicate a 
true north and south line. The slightest deviation from this 
position will eause the image to pass above or below the lines, 
and thus discover the error. 

We thus, from the position of the sun in the solar system, 
obtain a certain direction absolutely unchangeable, from which 
to run our lines, and measure the horizontal angles required. 

This simple principle is not only the basis of the construction 
of the Solar Compass, but the sole cause of its superiority to 
the ordinary or magnetic instrument. 

The sights and the graduated limb being adjusted to the 
solar apparatus, and the latitude of the place, and the declina¬ 
tion of the sun also set off upon the respective arcs, we are 
able not only to run the true meridian, or a due east and west 
course, but also to set off the horizontal angles with minuteness 
and accuracy from a direction which never changes, and is 
unaffected by attraction of any kind. 

Adjustments :— 

The adjustments of this instrument, with which the surveyor 
will have to do, are simple and few in number, and will now 
be given in order. 

% 

1. To adjust the Equatorial Lines and Solar Lenses. 

First detach the arm, A, from the declination arc by with¬ 
drawing the screws shown in the cut from the ends of the posts 
of the tangent -screw, A, and also the-clamp screw, and the 
conical pivot with its small screws, by which the arm and 
declination arc are connected. 

The arm, A, being thus removed, attach the adjuster in its 
place by replacing the conical pivot and screws, and insert the 
clamp screw so as to clamp the adjuster at any point on the 
declination arc. 

Now level the instrument, place the arm, A, on the adjuster, 
with the same side resting against the surface of the declination 
arc as before it was detached. Turn the instrument on its 
spindle so as to bring the solar lens to be adjusted in the direc* 



18 


SURVEYING AND NAVIGATION. 


tion of* the sun, and raise or lower tlie adjuster on the declina¬ 
tion arc, until it can be clamped in such a position as to bring 
the sun’s image, as near as may be, between the equatorial lines 
on the opposite silver plate; and bring the image precisely into 
position by the tangent screw of the latitude arc, or the level¬ 
ing screws of the tripod. Then carefully turn the arm half way 
over, until it rests upon the adjuster by the opposite faces of 
tiie rectangular blocks, and again observe the position of the 
sun’s image. 

If it remains between the lines as before, the lens and plate 
are in adjustment; if not, loosen the three screws which con¬ 
fine the plate to the block, and move the plate under their 
heads, until one half the error in the position of the sun’s 
image is removed. 

Agam bring the image between the lines, and repeat the 
operation until it will remain in the same situation in both 
positions of the arm, when the adjustment will be completed. 

To adjust the other lens and plate, reverse the arm end for 
end on the adjuster, and proceed precisely as in the former 
case until the same result is attained. 

In tightening the screws over the silver plate, care must be 
taken not to move the plate. 

This adjustment now being complete, the adjuster should 
be removed, and the arm, />, with its attachments, replaced 
as before. 

2. To adjust the Vernier of the Declination Arc. 

Having leveled the instrument, and turned its lens In the 
direction of the sun, clamp to the spindle, and set the vernier, 
v , of the declination arc, at zero, by means of the tangent 
screw at k , and clamp to the arc. 

See that the spindle moves easily and yet truly in the socket, 
or polar axis, and raise or lower the latitude arc by turning 
the tangent screw, f until the sun’s image is brought between 
the equatorial lines on one of the plates. Clamp the latitude 
arc by the screw, and bring the image precisely into position 


INSTRUMENTS. 


19 


by the leveling screws of tlie tripod or socket, and without 
disturbing the instrument, carefully revolve the arm, A, until 
the opposite lens and plate are brought in the direction of the 
sun, and note if the sun’s image comes between the lines a$ 


before. 

If it does, there is no index error of the declination arc; if 
not, with the tangent screw, A, move the arm until the sun’s 
image passes over half the error; again bring the image 
between the lines, and repeat the operation as before, until the 
image will occupy the same position on both the plates. 

We shall now find, however, that the zero marks on the arc 
and the vernier do not correspond ; and to remedy this error, 
the little flat head screws above the vernier must he loosened 
until it can be moved so as to make the zeros coincide, when 
the operation will be completed. 


3. To adjust the Solar Apparatus to the Compass Sights. 

First level the instrument, and with the clamp and tangent 
screws set the main plate at 90° by the verniers of the hori¬ 
zontal limb. Then remove the clamp screw, and raise the 
latitude arc until the polar axis is, by estimation, very nearly 
horizontal, and if necessary, tighten the screws on the pivots 
of the arc, so as to retain it in this position. 

Fix the vernier of the declination arc at zero, and direct the 
equatorial sights to some distant and well marked object, and 
observe the same through the compass sights. If the same 
object is seen through both, and the verniers read to 90° on 
the limb, the adjustment is complete; if not, the correction 
must be made by moving the sights, or changing the position 
of the verniers. 


HOW TO USE THE SOLAR COMPASS. 

Before this instrument can be used at any given place, it is 
necessary to set off upon its arcs both the declination of the 
sun as affected by its meridional refraction for the given day, 
and the latitude of the place where the observation is made. 


20 


SURVEYING AND NAVIGATION. 


1. To set off the Declination. 

The declination of the sun, given in the Ephemeris of the 
Nautical Almanac from year to year, is calculated for apparent 
noon at Greenwich, England. 

To determine it for any other hour at a place in the United 
States, reference must be had, not only to the difference of 
time arising from the longitude, but also to the change of de¬ 
clination from day to day. 

The longitude of the place, and therefore its difference in 
time, if not given directly in the tables of the Almanac, can be 
ascertained very nearly by reference to that of other places given, 
which are situated on, or very nearly on, the same meridian. 

It is the practice of surveyors in the States east of the Mis¬ 
sissippi, to allow a difference of six hours for the difference in 
longitude, calling the declination given in the Almanac for 
12 M., that of 6 A. M. at the place of observation. 

Beyond the parallel of Santa Fe the allowance would be 
about seven hours ; and in California, Oregon, and Washington 
Territory, about eight hours. 

Having thus the difference of time, we very readily obtain 
the declination for a certain hour in the mornino;, which would 
be earlier or later as the longitude was greater or less, and 
the same as that of apparent noon at Greenwich on the 
given day. » 

To obtain the declination for the other hours of the day, 
take from the Almanac the declination for apparent noon of 
the given day, and also that of the day following, subtract 
one from the other, as it may have increased or decreased, and 
we have the change of declination for 21 hours; divide this 
by 24, and we obtain the change of declination for a single 
hour, which is to be added to, or subtracted from, that of the 
starting hour, according as the declination is increasing or 
decreasing between the two days taken. 

To make this more plain, we will give an example. Sup¬ 
pose it was required to obtain the declination for the different 
hours of April 16th, 1863, at Troy, N. Y. 


INSTRUMENTS. 


21 


The longitude in time is 4 hours 54 minutes 40 seconds, or 
practically 5 hours, so that the declination given in the Al¬ 
manac, for the given day at Greenwich, would be that of 7 A.M. 
at Troy. To obtain the hourly change, 

Say declination at Greenwich, April 17, 10° 23 f 5S ,f 

“ “ “ 16, 10. 2 44 

tiff __ 

Change for 24 hours .... 21 14 

Reduce to seconds, and divide by 24, and we have an hour¬ 
ly change of 53 seconds, which, as the declination is increasing, 
is to be added every hour after 7 A. M, 

Hence, sun’s declination at Greenwich noon, as by the table, 


being 

p • p p • p * • 

10 

2 

44 





Add me: 

ridional refraction 



38 







10 

3 

22 Dec. foi 

’ 7 

A. M. 

Add hourly change . . , . 



53 







10 

4 

15 

a 

8 


(6 


u 

• • • • 



53 







10 

5 

8 

jU 

9 


a 

a 

t 0 • • 



53 







10 

6 

1 

a 

10 


(( 

a 

a 

• • • • 



53 







10 

6 

54 

a 

11 


u 

« 

u 

• P jP • 



53 







10 

7 

47 

u 

12 

M. 

» 

a 

*6 

• • • ,# 



53 







40 

8 

40 

u 

1 

P. 

M. 

a 

a 

• • p • 


- 

53 






' 

10 

9 

33 

u 

2 


a 


u 

• • • m 



53 







id 

10 

26 

.a 

3 


a 












22 


SURVEYING AND NAVIGATION. 


10 10 26 Dec*, for 3 P. M. 
Add hourly change.... 53 




10 

11 

19 

a 

4 

a 


a 

• • • • 



53 






10 

12 

12 


5 

a 


u 

• • • • 



53 





* 

10 

13 

5 

u 

6 

a 

a 

• • • • 



53 






10 

13 

58 

u 

7 

a 


In trie case taken, the declination is increasing from day to 
day, and therefore the hourly change is added ; if, on the con¬ 
trary, the declination was decreasing, the hourly change should 
be subtracted. 

The calculation of the declination for the different hours of 
the day should, of course, be made and noted, before the sur¬ 
veyor commences his work, that he may lay off the change 
from hour to hour, from a table prepared as above described. 

It is considered sufficiently accurate by most government 
surveyors, to set off the declination only three or four times in 
the day, at intervals of two or three hours as required. 

i } 

2. To set off the Latitude. 

Find the declination of the sun for the given day at noon afc 
the place of observation, as just described, and with the tangent 
screw set it off upon the declination arc, and clamp the arm 
firmly to the arc. 

Observe in the Almanac the equation of time to* the given 
day, in order to know about the time the sun will reach the 
meridian. 

Then, about fifteen or twenty minutes before this time, set 
up the instrument, level it carefully, fix the divided surface ot 
the declination arc at 12 on the hour circle, and turn the in¬ 
strument upon its spindle until the solar lens is brought into 
the direction of the sun. 






INSTRUMENTS. 


23 


Loosen tlie clamp screw of tlie latitude arc, and with the 
tangent screw raise or lower this arc until the image of the 
sun is brought precisely between the equatorial lines, and turn 
the instrument from time to time, so as to keep the image also 
between the hour lines on the plate. 

As the sun ascends, its image will move below the lines, and 
the arc must be moved to follow it. Continue thus, keeping it 
between the two sets of lines until its image begins to pass 
above the equatorial lines, which is also the moment of its 
passing the meridian. 

Now read off the vernier of the arc, and we have the lati¬ 
tude of the place, which is always to be set off on the arc when 
the compass is used at the given place. 

3. To Run Lines with the Solar Comjpass. 

Having set off, in the manner just given, the latitude and 
declination upon their respective arcs, the instrument being 
also in adjustment, the surveyor is ready to run lines by 
the sun. 

To do this, the instrument is set over the station and care¬ 
fully leveled, the plates clamped at zero on the horizontal limb, 
and the sights directed north and south, the direction being 
given, when unknown, approximately by the needle. 

The solar lens is then turned to the sun, and with one hand 
on the instrument, and the other on the revolving arm, both 
are moved from side to side, until the sun’s image is made to 
appear on tlie silver .plate; when, by carefully continuing the 
operation, it may be brought precisely between the equatorial 
lines. 

Allowance being now made for refraction, the line of sights 
will indicate the true meridian; the observation may now be 
made, and the flag-man put in position. 

When a due east and west line is to be run, the verniers of 
the horizontal limb are set at 90°, and the sun’s image kept 
between the lines as before. 


SURVEYING AND NAVIGATION. 


21 

USE OF THE NEEDLE. 

In running lines, tire magnetic needle is always kept with 
the sun; that is, the point of the needle is made to indicate 0 
on the arc of the compass box, by turning the tangent screw 
connected with its arm on the opposite side ot the plate. By 
this means, the lines can be run by the needle alone, in case 
of the temporary disappearance of the sun; but in such cases, 
of course, the surveyor must be sure that no local attraction is 
exerted. 

The variation of the needle, which is noted at every station, 
is read off in degrees and minutes on the arc, by the edge ot 
which the vernier of the needle box moves. 

I 

TIME OF DAY BY THE SUN. 

The time of day is best ascertained by the Solar Compass 
when the sun is on the meridian, as at the time of making the 
observation for latitude. 

The time thus given is that of apparent noon, and can be 
reduced to mean time by merely applying the equation of time 
as directed in the Almanac, and adding or subtracting as the 
sun is slow or fast. 

ADVANTAGES OF THE SOLAR COMPASS IN SURVEYING. 

It will readily occur to all who have read the preceding 
description of the Solar Compass, that while it is indispensable 
in the surveys of public lands, it also possesses important ad' 
vantages over the magnetic compass when used in the ordi¬ 
nary surveys of farms, &c. 

For not only can lines be run and angles be measured with¬ 
out regard to the diurnal variation, or the effect of local 
attraction, but the bearings, being taken from the true meri¬ 
dian, will remain unchanged for all time. 

The constant uncertainty caused by the variation of the 
needle, and the litigation to which it so often gives rise, may 
thus be entirely prevented by the use of the Solar Compass in 
this kind of work. 


i 











% 



fN 



















INSTRUMENTS. 


25 


THE TRANSIT. 

The Transit is an instrument designed for measuring either 
horizontal or vertical angles. It consists of a telescope re¬ 
volving by its cross bar, in the top of standards attached to a 
horizontal vernier plate, that moves around a graduated circl 
;>r limb beneath. 

The telescope “ transits ,” or turns, completely around in a 
vertical plane, so as to be directed to opposite points without 
revolving the instrument. Within the tube of the telescope, 
and in the exact focus of the object and eye-glasses, is a small 
ring, upon which are stretched two fine spider lines at right 
angles to each other, and forming by their intersection a 
minute point, by which the telescope can be directed to any 
object desired. 

The vertical circle is connected with the axis of the tele¬ 
scope, and turns with it. 

Both the horizontal and vertical circles are divided to halt- 
degrees, and read by their verniers to single minutes. 

Clamp and tangent screws are also connected with both the 
horizontal and vertical circles, by which they are slowly 
moved a short distance in either direction, and set with the 
utmost nicety to any position desired. 

A compass circle with a magnetic needle is always connected 
with the Transit, and is used precisely like that of the Sur¬ 
veyor’s Compass already described. 

When this circle is fixed to the vernier plate, the instru¬ 
ment is termed an Engineer s Transit • when it can be turned, 
like that of the compass, to set off the variation of the needle, 
as represented in the engraving, it is called a Surveyor’s 
Transit . 

A level, as shown in the engraving, is often placed on the 
under side of the telescope, and is used to determine a hori¬ 
zontal line. 


2 


i 


26 SURVEYING AND NAVIGATION. 
Adjustments :— 

1. The intersection of the spider lines must be in the optical 
axis of the telescope, so that the instrument, when placed in 
the middle of a straight line, will, by the revolution ot the 
telescope, cut its extremities. 

To do this, set up the instrument firmly, level the plates, 
and having clamped t\\ firmly together, direct the telescope 
by the cross wires to some distant object; revolve the telescope, 
and find an object in the opposite direction, which the cross¬ 
wires will bisect; the two objects would be exactly in line, if 
the telescope was in adjustment. 

To test this, turn the instrument half way around, set the 
cross-wires precisely upon the second object found, and again 
revolve the telescope in the direction of the first object. 

If the cross-wires strike it, the telescope is in adjustment; 
if not, by the capstan head screws, shown near the cross-bar, 
move the ring until, by estimation, one-fourth the error is cor¬ 
rected. 

Repeat the operation until the adjustment is complete, and 
the cross-wires will cut the ends of a straight line. 

2 . The vertical circle, with its vernier, must be adjusted to 
the horizontal cross-wire of the telescope. 

To do this, level the instrument carefully, and set the zeros 
of the circle and vernier in exact coincidence; and have the 
clamp to the telescope axis firmly fastened. 

Find or set some distant object, which the horizontal cress- 

% 

wire will precisely indicate; unclamp the telescope axis, turn 
the instrument halfway around, revolve the telescope, and set 
the horizontal wire upon the same point; and note whether 
the zeros of the vertical circle and its vernier correspond as 
before. If not, slacken the two screws which confine the ver¬ 
nier, and move it until half the error is removed. 

Repeat the same operation until the adjustment is complete. 

The other adjustments of the transit are mainly the same as 
those of the compass already given. 


INSTRUMENTS. 


27 


/ 


1. To measure horizontal angles with the transit. 

Set tlie instrument, by a suspended plummet, directly over 
the point ot observation, level the plates by means of the large 
screws underneath, bring the telescope upon one of the points 
to be observed, and note the exact reading of tlie horizontal 
circle or limb. 

Then unclamp the horizontal plates, and turn the telescope 
so as to strike the second object selected; note the reading of 
the verniers of the limb, and the difference between the two 
readings w r ill be the angle required. 

2. To measure vertical angles with the transit. 

If the instrument is in adjustment, direct the telescope to 
any object whose altitude is required, and read the angle by 
the vertical circle and vernier; this will be the altitude re 
quired. 

3. To run levels with the transit. 

Level the instrument carefully, bring the telescope into a 
horizontal position, clamp the axis, and with the tangent 
screw bring the level bubble under the telescope precisely 
into the centre, and the horizontal cross-wire will indicate a 
level line in any direction. 

THE SURVEYOR’S CHAIN. 

In farm surveying, lines are measured with a chain 66 feet, or 
4 rods long. This chain is divided into 100 links, each being 
.92 inches long; 10 square chains make an acre, and as the 
links are hundredths of a chain, we can say, 2 chains and 46 
links, or 2.46 chains, or 246 links. And the area computed in 
chains can be expressed in acres by moving the decimal point 
one place to the left. To facilitate the counting, every tenth 
link is indicated by a small piece of brass. 

Note.— Engineers measure lines with a chain 100 feet long. This is 
divided into 100 links, each link being one foot long. 


28 


SURVEYING AND NAVIGATION. 


PLANE TABLE. 

The Plane Table is exactly what the name indicates; it is 
a plane hoard table, about two feet long and 20 inches wide, 
resting on a tripod, to which it is firmly screwed, yet capable 
of an easy motion on its centre, having a ball and socket like 
a compass staff. 

Directly under the table is a brass plate, in which four 
milled screws are worked, for the purpose ot adjusting the 
table, the screws pressing against the table. 

To level the table, a small detached spirit-level may be used. 
The level being placed on the table over two of the screws, the 
screws are turned contrary ways until the level is horizontal; 
after which it is placed over the other two screws, and made 
horizontal in the same manner. 

The table has a clamp screw to hold it firmly during obser¬ 
vations, and also a tangent screw to turn it minutely and 
gently, after the manner of the theodolite. 

The upper side of the table is bordered by four narrow brass 
plates, and the centre of the table is marked by a pin. 

About this center, and tangent to the corners of the table, 
conceive a circle to be described. Suppose the circumference 
of this circle to be divided into degrees and parts of a degree, 
and radii to be drawn through the center, and each point of 
division. 

The points in which these radii intersect the outer edge of 
the brass border, are marked by lines on the brass plates ; these 
lines, of course, show degrees and parts of degrees; they ai * 
marked from right to left, from 0 to 180° on both sides, but on 
some tables the numbers run all the way round, from 0 to 360°. 

Near the two ends of the table are two grooves, into which 
are fitted brass plates, which are drawn down into their places 
by screws coming up from the under side. The object of these 
grooves and corresponding plates, is to hold down paper firmly 
and closely to the table. 

The paper, before being put on, should be moistened to ex- 


INSTRUMENTS. 


20 


pand it; then, carefully drawn over the table, and fastened 
down by the plates that fit into the grooves ; on drying, it will 
closely fit the table. 

A delicate fine-edged ruler is used with the plane table ; it 
has vertical sights, the lines of which are in the same vertical 
plane as the edge of the ruler. 

The plane table may be used for three distinct objects. 

1st. For the measurement of horizontal angles. 

2d. For the determination of the shorter lines of a survey, 
both as to extent and position. 

3d. For the purpose of mapping down localities, harbors, 
water courses, &c. 


1. To measure a horizontal angle. 

Place the center of the table over the angular point. Level 
the table, then place the edge of the ruler against the pin at 
the center; direct the sights to one object, and note the degree 
on the brass plate ; then turn the ruler to the other object, and 
note the degree as before. The difference of the degrees thus 
noted is the angle sought. 

If the ruler passed over 0 in turning from one object to the 
other, subtract the larger angle from 180°, and to the remain¬ 
der add the smaller angle. 



2. To determine lines in extent and position. 

Let CD, in the diagram, be a base 
line; place the table over C so that 
the point on the table, where we 
wish C to be represented, shall fall 
directly over 6 7 / and place the table 
in such position that CD shall take 
the desired direction on the table. 

How level the instrument, and 
clamp it fast; it is then ready for use. 

Sight to the other end of the base line, and mark it along 
the fine edge of the ruler. 






V 


I 

30 SURVEYING- AND NAVIGATION. 

In the same manner, sight along the direction of CE, and 
mark that direction in a fine lead line, that can be easily 
rubbed out; the point E is somewhere in that line. 

Sight in the direction of E, and mark the line on the paper; 
F is somewhere in that line. In this manner, sight to as many 
objects as desired, as G, II , B, A, &c. 

Now the base on the paper, may be as long, or as short r.3 
we please; suppose the real base on the ground to be 1,200 
feet; this may be represented on the table by 3, 4, 5, 10, or 
12 inches, more or less. Suppose we represent it by 6 inches; 
then one inch on the paper will correspond with 200 feet on 
the ground ( horizontally ). 

Take CD, six inches, and place a pin at D, remove the in¬ 
strument to the other end of the base, and place D of the table 
right over the end of the base, by the aid of a plumb, and 
give the table such a position as will cause CD on the table, 
to correspond with the direction of the base. 

Level the table and clamp it. Now, if CD on the table 
does not exactly correspond with the direction of the base on 
the ground, make it correspond by means of the tangent 
screw. 

Now from D, by means of the ruler and its sight vanes, 
draw lines on the paper, in the direction of the points E, F, G, 
II, B, A, &c. ; and where these lines intersect those from the 
other end of the base to the same points, are the real localities 
of those points, in proportion to the base line. Lines drawn 
from point to point, where these lines intersect, as EE, EG, 
GII, &c., will determine the relative distances from point to 
point, at the rate of 200 feet to the inch. 

Lines drawn from the center of the table, parallel to EE 
and EG, will determine the angle EFG, in case the angle is 
required. After the points, E, F, G, &c., have been deter¬ 
mined, the light pencil lines to them, from the ends of the 
base, may be rubbed out, except those that we may wish to 
retain. 


















«r 












INSTRUMENTS. 


31 


THE ENGINEER’S LEVEL. 

V 

The engineer’s level is an instrument by which we can mark 
a horizontal line, or a line parallel to the surface of tranquil 
water, and by which we can ascertain how much certain 
points are above or below the line marked out. This instr- 
lnent consists essentially of a telescope and an attached level. 

Adjustments :— 

1. The intersection of the spider lines should be in the 
optical axis of the telescope. 

Direct the telescope to some distant object, and bring the 
intersection of the spider lines upon some distinct point; 
then revolve the telescope on its bearings. If the intersection 
moves from the point, change the position of the spider lines 
by the screws attached, until the intersection will remain 
upon the same point while the telescope is revolved upon its 
bearings. 

2 . The level attached should be parallel to the optical axis 
of the telescope. 

Brins: the bubble to the middle of the level tube : then take 
the telescope from its bearings, and reverse it. If the bubble 
does not remain at the middle, the tube must be changed in 
position by the screws at its end, until the bubble will settle 
at the middle after the telescope is reversed. 

3 . The optical axis of the telescope should be perpendicular 
to the axis of the instrument, so that the bubble will remain 
in the middle of its tube during an entire revolution of the 
instrument on its axis. Bring the telescope directly over two 
opposite leveling screws ; then bring the bubble to the middle 
of its tube, and revolve the instrument on its vertical axis ; if 
the bubble moves from the middle, one of the standards or 
wves must be moved by the nuts on either side of the bar until 
the instrument can be revolved without displacing the bubble. 

The Tripod Head of the engineer's level has two parallel 
plates connected by a ball and socket-joint, upon which the 


/ 


32 


SURVEYING AND NAVIGATION. 


upper plate moves. There are four large-headed screws called 
leveling screws, which move in bearings in the upper plate, 
and which, by pressing upon the lower plate, keep the two 
plates firmly apart. 

To level the instrument, turn the telescope directly over tw T o 
opposite leveling screws. Then turn these two screws in 
contrary directions moving both thumbs in or out as may be 
required until the bubble is in place. Then place the teles¬ 
cope directly over the other opposite leveling screws, and turn 
these two in contrary directions, until the bubble is in place. 

There are also clamp and tangent screws, giving a slow 
motion to the telescope, to enable the observer to bring the 
spider lines precisely upon any point. The use of these will 
be readily understood from an examination of the instrument. 

To Test tlie Adjustment of the Level. 

It is important that the level should be in as perfect adjust¬ 
ment as possible. The accuracy of the adjustments may be 
tested as follows. 



Measure very carefully the distance between two stations, 
as E and F\ and set the instrument exactly midway between 
them as represented in the last figure. 

Then level the instrument, and find the difference of the 
>evels between if and F( two pegs driven into the ground). 

Now, suppose A E measures on the rod, . 4.752 feet. 

And BE “ “ “ . 6.327 feet. 

Then E is above F . 


. 1.575 feet. 










INSTRUMENTS. 


33 


The Engineer’s Leveling Rod is formed of 
two pieces of wood sliding from each other, and 
graduated into feet, tenths, and hundreds, 
with verniers reading to thousandths of a foot. 
The target is circular, and painted white and 
red in alternate quadrants. It is kept in place 
by a clamp-screw, and it carries upon the front 
side of the rod a vernier reading to the thou¬ 
sandth of a foot. The front surface on which 
the target moves, reads to six and a half feet; 
when a greater height is required, the hori¬ 
zontal line of the target is fixed at that point, 
and the upper half of the rod carrying the 
target is moved out of the lower, the read¬ 
ing being now obtained by a vernier on the 
graduated side, up to an elevation of nearly 
twelve feet. 

The verniers on the engineer’s leveling rod 


Now bring the level near to one of the stations as E\ and 
level it very accurately, and sight to the rod A. E. 

Now, suppose the target stands at 5.137 feet. 

To this add ... 1.575 feet. 

b7fl2 

The rod man now goes to the station F T , puts 
his target on the rod exactly at 6.712, and the 
telescope is turned upon it, and the horizontal 
spider line ought to just coincide with the 
target, and will, if the instrument is in perfect 
adjustment; if it is not, the error is taken out 
by the screws shown on the outside of the tube. 

It the error was but slight, as in such cases it 
always is, with good instruments, the adjust¬ 
ment is as complete as it can be made. 


THE LEVELING ROD. 


m 
























































































✓ 


34 SURYEYING AND NAVIGATION. 

✓ 

are so constructed that ten spaces of the vernier scale are just 
equal to nine of the smallest spaces on the graduated rod ; 
and as these last spaces are hundredths of a foot, the verniers 
will read to the thousandth of a foot. 


THE THEODOLITE. 


The Theodolite is an instrument designed for the measure¬ 
ment of either horizontal or vertical angles. It answers the 
same purposes, therefore, in the main, as the transit. Like 
the transit, it has two graduated circles, one horizontal, and 
the other vertical; also a telescope, and generally a compass 
box. The telescope, however, rests in Y shaped supports, as 
in the engineer’s level, and may he reversed without dis¬ 
turbing the position cf the vertical circle with which it is 
connected. 


Adjustments :— 

1. The intersection of the spider lines should be in the opti¬ 
cal axis of the telescope. 

2 . The level attached to the telescope should be parallel to 
the optical axis of the telescope. 

3 . The axis of the telescope should be parallel to the hori¬ 
zontal plate. 

4 . The optical axis of the telescope should be perpendicular 
to the axis of motion. 

5 . The vertical circle should be perpendicular to the axis 
of the telescope and the horizontal plate. 

The first two of these adjustments are made precisely as the 
corresponding adjustments of the level. The other three are 
usually attended to by the maker. 

The theodolite is less convenient in practical field work 
than the transit, and has never been popular in our own 
country. 


INSTRUMENTS. 

SECTION II. 

INSTRUMENTS IN NAVIGATION. 


THE SEXTANT. 



The Sextant is an instrument much used at sea for measur¬ 
ing angles; its construction is such that observations made 
with it are not essentially affected by the motion of the ship. I 
In the figure above, which represents the sextant, BC is a 
graduated arc, a little more than sixty degrees in extent; the 
graduation commences at B. The graduated arc is firmly 
connected with a frame of brass or ebony. A Vis a revolving 
index bar , turning on a pivot at the center of the circular part 
at A , this point being also the center of the graduated arc. 
Attached to the index bar, near V, is a vernier scale , which 















36 


SURVEYING AND NAVIGATION. 


slides over the graduated arc, and which may be fastened at 
any point by a clamp screw, or moved gently by a tangent 
screw. There is also a microscope attached to the bar, which 
may be turned over the vernier, and used to distinguish the 
lines of graduation. At A is a small plane mirror, perpendi¬ 
cular to the plane of the sector; it is attached to the index 
bar, and revolves with it. This is called the index glass « 
Attached to the frame of the instrument, at AT, is another 
mirror, called the horizon glass. It is fixed at right angles 
with the plane of the instrument, and is parallel to the index 
glass when the index is at zero of the arc. The lower half of 
the horizon glass is silvered, so as to make it a reflector; the 
upper half is transparent. There is a small telescope, E y 
which is fixed in a ring, and directed towards the horizon 
glass at II. At M and N are sets of colored glasses or 
screens, one or more of which may be thrown between the 
two mirrors, or between the horizon glass and telescope, to 
moderate the light of the sun or moon, when under obser¬ 
vation. 

The sextant when used is held by the handle at P , so that 
the plane of the graduated arc shall coincide with the two 
points whose angular distance is to be measured. The tele¬ 
scope is directed to one of the points or bodies, which is seen 
directly through the upper or transparent part of the horizon 
glass. The index bar is then turned, until the other point or 
body, whose image is reflected from the index glass to the 
horizon glass and thence into the telescope, comes into ap¬ 
parent contact with the first. The angular distance of the two 
points is then indicated by the vernier. 

The angular distance measured between any two objects is 
always twice the number of degrees over which the index 
bar moves. 

To show this, we will make use of a diagram, showing onlv 
the lines of direction assumed by the principal parts of the 
instrument. A V is the direction of the plane of the index 


INSTRUMENTS. 


37 


glass, and IIL the direc¬ 
tion of the plane of the 
horizon glass. The eye 
piece of the telescope is 
supposed to be at E. 

Now conceive a ray of 
ight coining from an ob¬ 
ject 8, and striking the 
mirror A , the index mir¬ 
ror being turned so as to 
throw the reflecting ray into 
the mirror II ‘ this mirror 
again reflects it towards A?, and an eye anywhere in the line Dll 
will see the image of the object behind the mirror II. Conceive 
the ray of light from S to pass right through the mirror at A, 
to meet the line IIE; then, it is obvious that the angle SED 
measures the angle between the object S and its image D. 

It is a principle in optics that the angle of incidence is ecpial 
to the angle of reflection ; from this it follows that VA bisects 
the angle EAII in the diagram ; and III, the direction of the 
horizon glass, bisects the angle EHC. In the triangle EAII, 
the angle at E is equal to the difference of the exterior angle 
EHC, and the interior angle EAII. 

Also, in the triangle I AH, the angle at* L is equal to the 
difference of the exterior angle LI1C, and the interior angle 
LAI1. Therefore we have 

E = EHC - EAII, 

L = LHC- LAII. 

But EHC is twice LI1C, and EAII is twice LAI1 • whence it 
follows that the angle E is twice the angle L ; but the angle 
L is equal to the angle BA V, because IIL is parallel to AB. 
Therefore the angle at E is twice the angle BA V, which is 
measured by the arc B V, over which the zero of the index 
arm has moved. It is obvious that double the number of 
degrees in the arc B V will be the number of degrees in the 
angle E. 











38 


SURVEYING AND NAVIGATION. 


For convenience, however, every half degree of the gradu* 
ated arc is reckoned as a degree; for it represents a degree ot 
the angle measured. In the figure representing the sextant* 
the arc J3C is sixty-five degrees in extent, but the 130 halt 
degrees into which it is divided are reckoned as 130 degrees. 


Adjustments: . 

1. The index glass should be perpendicular to the plane of 
the instrument. 

Place the index near the middle of the arc, and look into 
the index glass in the direction of the plane of the instrument. 
If the reflected arc appear in a line with the arc seen direct, 
the index glass is perpendicular to the plane of the instru¬ 
ment ; if not, the index glass must be moved until the arc and 
its image appear in a line. 

2 . The horizon glass should be parallel to the index glass 
when the index is placed at zero. 

Clamp the index at zero, hold the instrument vertically, and 
see if the distant horizon coincides with its image, as seen in 
the horizon glass, so as to form one continued line; if not, the 
horizon glass must be moved by its screws until the object and 
its image coincide. 

3 . The horizon glass should be perpendicular to the piano 
of the instrument. 

Clamp the index at zero, and look at some smooth portion 
of the distant horizon while holding the instrument perpendic¬ 
ular; a continued unbroken line will be seen in both parts of 
the horizon glass; and if, on turning the instrument from the 
perpendicular, the horizontal line continues unbroken , the hori¬ 
zon glass is in full adjustment; but if a break in the line is 
observed, the glass is not perpendicular to the plane of the in¬ 
strument, and must be made so by the screw adapted to that 
purpose. 

After an instrument has been examined according to these 
directions, it may be considered as in an approximate adjust¬ 
ment ; a re-examination will render it more perfect. 




INSTRUMENTS. 39 

i 

Finally, we may find its index error as follows: Measure 
the sun’s diameter both on and off the arch—that is, both ways 
from 0; and if it measures the same, there is no index error. 
But if there is a difference, half that difference will be the in¬ 
dex error, additive , if the greater measure is off the arch, sub- 
tractive , if on the arch. 

1. To measure the altitude of the sun at sea. 

Turn down the proper screen or screens to defend the eye. 
Put the index at 0, having it loose, look directly at the sun 
through the tube, and you will see its image in the silvered 
part of the horizon glass. Now move the index, and the image 
will drop; drop it to the horizon, and clamp the index. 

Let the instrument slightly vibrate each side of the perpen¬ 
dicular, on the line of sight as a center, and the image of the 
sun will apparently sweep along the horizon in a circle. While 
thus sweeping, move the tangent screw, so that the lower limb 
of the sun will just touch the horizon, without going below it. 
The reading of the index will be the altitude corresponding to 
that instant, provided there be no index error. 

2. To measure the angular distance between two bodies as 
the sun or moon , or the moon and a star. 

The most brilliant of the two objects is always reflected to 
the other. Loosen the index, place it at 0, and direct the line 
of sight to the brighter object, and catch a view of its image 
in the silvered part of the horizon glass. 

Turn the plane of the instrument into the plane between the 
two objects; now move the index, keeping the eye on the 
image, and bring it along to the other object; bring them as 
near as possible, then gently clamp the index. 

Hold up the instrument again, in the plane between the 
two objects, and view one object through the transparent part 
of the horizon glass; and when the instrument is in the right 
position, the image of the other object will appear also in the 
same field of view, and then, with the tangent screw, make the 


I 


40 SURVEYING AND NAVIGATION. 

limb of the reflected object just touch the other, as it moves 
past it, to and fro, by the gentle motion of the instrument, 
until the observer is satisfied that he has got the measure as 
near as he can. 

The limb of the sextant is divided into degrees, and each 
degree subdivided into six equal parts by short lines, each of 
hese parts being read as ten minutes. 

The vernier is so constructed that 60 spaces on the vernier 
scale are just equal to 59 of the smallest spaces on the 
limb ; and as these spaces are 10' each, the vernier will read 
to 10". 

Now, if the zero mark of the vernier coincides with any line 
on the limb, then that line will indicate the angle. Thus, if 
it coincides with the line marked 30, then the angle is 30°. 
If the zero coincides with the next long mark beyond that 
marked 30, then the angle is 31°. If the zero does not coin¬ 
cide with the next long line, but is beyond it, and coincides 
with one of the shortest lines, then the angle will be 31° and 
so many minutes, counting each small space as 10'. When 
the zero division of the vernier does not coincide with any line 
upon the limb, but stands between two of them, then look 
along the vernier scale for a line that does coincide with a line 
on the limb; then count from the zero to the coinciding line 
upon the vernier, calling each small space of the vernier 10"; 
the result added to the reading of the limb will give the angle. 
If, for example, the zero of the vernier is between the second 
and third short lines, which are between the first and second 
long lines beyond 30 on the limb, and the fourth short line be¬ 
yond 3 on the vernier coincides with a line on the limb; then 
from the limb we read 31° 20', and from the vernier we read 
3 ; 40", and the two readings give for the angle 

31° 23' 40". 

In reading from any vernier, first ascertain the value of the 
lowest division of the limb of the instrument, and then find 
the relation of one division of the vernier to one division of 


INSTRUMENTS. 


41 


the limb. This can be done by finding how many of the 
divisions of the limb are equal to the graduated part of the 
vernier. 

THE LOG. 

The rate at which a ship sails is measured by a line running 
oF of a reel, called the log line. 


The log is nothing more than a piece of thin board in the 
form of a sector, of about six inches radius : the circular part 
is loaded with lead to make it stand perpendicular in the water. 

The line is so attached to it that the flat side of the log is 
kept toward the ship, that the resistance of the water against 
the face of the log may prevent it, as much as possible, from 
being dragged after the ship by the weight of the line or the 
friction of the reel. 

The time which is usually occupied in determining a ship’s 
rate is half a minute, and the experiment for the purpose is 
generally made at the end of every hour, but in common 
merchantmen at the end of every second hour. As the time 
of operating is half a minute, or the hundred and twentieth 
part of an hour, if the line were divided into 120tlis of a 
nautical mile, whatever number of those parts a ship might 
run in a half minute, she would, at the same rate of sailing, 
run exactly a like number of miles in an hour. The 120th 
part of a mile is by seamen called a knot, and the knot is 
generally subdivided into smaller parts, called fathoms. 
Sometimes (and it is the most convenient method of division) 
the knot is divided into ten parts, more frequently perhaps 
into eight; but in either case the subdivision is called a 
fathom. 

The sixtieth part of a degree is called a nautical mile . 




























42 


SURVEYING AND NAVIGATION. 


We slmll consider a fathom the tenth of a knot, and as the 
nautical mile is 6,079 feet, the 120th part ot it is 50.66, the 
length, of a knot on the line, and a little over five feet is the 
length of a fathom. 

The operation of ascertaining the rate of sailing is called by 
-eainen heaving the log. 

At the end of an hour the loaded chip , or log, is thrown 
over the stern into the sea; a quantity of the line, called the 
stray line , is allowed to run off, then the glass is turned, and 
the number of knots that runs off the reel during the half 
minute is the rate of the ship’s motion. 

The log is then hauled in, and the same operation is repeated 
at the end of the next hour. 

The officer of the watch, who has been on deck during the 
hour, will mark on the slate or board, called the log board, 
the number of miles and parts of a mile which the ship has 
sailed during the last hour, according to the best of his judg 
merit / the log was thrown only to help make up that judg¬ 
ment, for the rate at the time the log was thrown may have 
been considerably more or less than the average motion during 
the hour. 


SECTION III. 

PLOTTING INSTRUMENTS. 

Plotting Instruments are those instruments used in de¬ 
lineating upon paper the lines of any survey, in their relative 
position and true proportion. 

The following are instruments which require no extended 
description. 

1. The Proportional Compass is an instrument employed 
for laying off' lines proportional to given lines. It can be so 
adjusted, by moving the pivot, as to secure any required ratio 
between the lines to be drawn and the given lines. {See Fig. 1.) 



/ 



2. The Dividers are used in describing arcs of circles with 
given radii, and in transferring proportional lines to paper, 
{See Fig 2.) 

3. The Parallel Rule is used in drawing lines parallel to 
a given line. {See Fig. 3.) 

4. The T Square is used in drawing perpendiculars and 
parallels. {See Fig. 4.) 

5. The Protractor is a graduated semi-circle used in meas¬ 
uring angles and in transferring them to paper. {See Fig. 5.) 

6. The Engineer’s Protractor is only another form of the 
instrument last described. It consists of a rectangular piece 
of ivory or metal, graduated upon three sides by lines converg¬ 
ing to a center in the fourth side. The graduations represent 
degrees, or parts of a degree, as in the other form. {See Fig. 6.) 

















































44 


/* 


SURVEYING AND NAVIGATION. 

7, The Pantograph is an instrument used for taking an 
enlarged or diminished copy of any figure, the exact form 
being preserved. It can be so adjusted that the lines of the 
copy shall bear any required ratio to the corresponding lines 
of the original. (See Fig. 7.) 

THE DIAGONAL SCALE OF EQUAL PARTS. 



This scale is used in transferring lines measured in units, tens 
and hundreds, or units, tenths and hundredths. 

To represent 2 chains and 46 links, place one foot of the 
dividers on the sixth parallel below 2, and extend the other 
foot to where the diagonal .4 intersects the parallel .06; and 
the space included will represent 2 chains 46 links on a scale 
of one chain to the inch, or 24 chains 60 links on a scale of ten 
chains to the inch, or 246 chains on a scale of one hundred 
chains to the inch. 


GUNTER’S SCALE. 

Gunter’s Scate is commonly two feet in length, containing 
the plane scale, and the scale of sines, chords, and tangents on 
one side of it, and the scale for the logarithms of numbers, 
sines, and tangents on the other. The logarithmic scale is not 
much used. 

The plane scale includes, in addition to the scale of equal parts, 
a line of chords formed by transferring the chords of the several 
arcs, into which a quadrant is divided, to a straight line, the 
distance from 0 to 60 being the radius of the quadrant; al&o. 


r 








































INSTRUMENTS. 


45 


a line of rhumbs, which is a line of chords where the quadrant 
is divided into 8 equal parts, called points. 

There are also lines of natural sines, tangents and secants, 
and a line of semi-tangents used in projections, as a line of longi¬ 
tudes used in navigation, and a line of latitudes used in the 
construction of sun-dials. 

The lines of equal parts are by far the most useful of tl e 
plane scale ; these lines are sometimes so graduated as to give 
10, 15, 20, 25, &c., parts to the inch. 

1. To lay down a given angle with the line of chords. 

Let it be required to con¬ 
struct an angle of 40° at the 
point A. With the dividers 
take 60 from the line of chords, 
and with one foot at A de¬ 
scribe the arc BG with 60 as 
radius; then from the same 
line of chords take 40, and with 
one foot of the dividers at B , mark the point D with the 
other foot, and draw the line AD. 

The angle BAD will be the required angle of 40°. 

2. To measure an angle with the line of chords. 

Let it be required to measure the angle at A in the diagram. 
With the chord of 60, and A as center, describe the arc BD , 
intersecting the sides that include the angle; then extend the 
dividers from B to D , and with that distance apply them to 
the line of chords, one foot being at the zero of the line; the 
other will point out the degrees in the angle. 

Angles are constructed and measured much more readily 
with a protractor than with the lines on the plane scale. 





) 


46 SURVEYING AND NAVIGATION. 

THE SECTOR. 


A 



% 

The Sector consists of two graduated arms, movable about 
a common point as center. On each arm are several lines 
diverging from the central point. One of these lines is divided 
into equal parts, and is called the line of lines; another is so 
graduated as to form a line of chords; another the line of 
sines ; another the line of tangents, &c. The lines of chords, 
sines, and tangents, are constructed upon the same radius, so 
that when the sector is open to any angle, the distance from 
60 to 60 on the line of chords is the same as the distance from 
90 to 90 on the line of sines, or the distance from 45 to 45 on 
the line of tangents. 

The principle of the sector depends upon the proportional¬ 
ity of the sides of similar triangles; and as the lines on one 
arm are graduated precisely the same as the corresponding 
lines on the other arm, it is obvious that when the sector is 
open at any angle, as in the diagram, we shall have— 

CA : AB:: CA': A'B'. 

Therefore A'B' is the same part of AB that CA is of CA. Hence 
if CA is a chord of any angle to CA as radius, A'B 1 will be 
a chord of the same angle to AB as radius; and if CA 1 is a 
sine of any angle to CA as radius, then A'B 1 will be a sine of 
the same angle to AB as radius. 

Examples. —1. To find the chord of 40° to a radius of 5 
inches, open the sector so that the distance from 60 to 60 on 
the line of chords shall be 5 inches; then the distance from 40 
to 40 on the same line will be the chord required. 




INSTRUMENTS. 


47 


2. To find the sine of 44° to a radius of 6 inches, open the 
sector so that the distance from 90 to 90 on the line of sines 
shall be 6 inches; then the distance from 44 to 44 on the line 
of sines will be the sine required. 

The sector is also used in drawing lines to any proposed 
scale. 

Example .—To draw r a line of 37 feet on a scale of 20 feet to 
tie inch; open the sector so that the distance from 20 to 20 
on the line of equal parts shall be one inch; then the distance 
from 37 to 37 on the same line will represent 37 feet on a scale 
of 20 feet to the inch. 

To divide a given line into any number, say 5 equal parts, 
open the sector so that the distance from 5 to 5 on the line 
of equal parts shall equal the given line; then the distance 
from 1 to 1 on the line of equal parts will be one of the re¬ 
quired parts. 

The advantage of the sector will appear from the following 
problem. 

A map is before me, its scale is 20 miles to an inch; I wish 
to find the distance in a right line between two points laid 
down on it. 

1st. I take one inch in the dividers, and open the sector so 
that the distance between 20 and 20 on the two arms, shall 
just correspond to the measure in the dividers, that is, shall 
be one inch. Let the sector lie on the table thus opened. 

2d. Now take the distance you wish to measure, in the 
dividers; place one foot on one arm of the sector, and the 
other foot on the other arm; so that the feet of the dividers 
s iall fall on the same number on both arms of the sector. 

The number thus marked by the dividers will be the dis¬ 
tance required. The distance between any two other points 
may be measured on the same map, without any computation 
whatever. 

It is obvious from the construction of the sector that such 
problems as finding third proportionals, fourth proportionals, 
or mean proportionals, can readily be solved from the line of 


48 


SURVEYING AND NAVIGATION. 


equal parts; and also that the various proportions in trigo¬ 
nometry can be worked by taking the sides of the triangles 
from the line of equal parts, and the degrees and minutes from 
the lines of sines, tangents 

On some sectors there are other lines, such as a line of poly¬ 
gons, a line of solids, etc. But these are more curious than usefu.. 


SECTION IV. 


PROBLEMS SOLVED INSTRUMENTALLY. 


With the instruments previously described, solve the fol¬ 
lowing problems. The references are to Robinson’s New 
Geometry. Thus, (th. 15, b. 1, cor. 1), indicates theorem 15, 
book 1, corollary 1, where the demonstrations of the problem 
referred to will be found. 

PROBLEM I. 


To bisect a given finite straight line. 

Let AB be the given line, and from 
its extremities, A and B , with any radius 
greater than the half of AB , describe 
arcs, cutting each other in n and m. A 
Join n and m; and 6 7 , where it cuts AB y 
will be the middle of the line required. 

Proof, (th. 18, b. 1, sell. 2). 




t> 


PROBLEM II. 

* 

To bisect a given angle. 

Let ABC be the given angle. With 
any radius, from the center B , describe 
the arc AC. From A and C, as centers, 
with a radius greater than the half of A C\ 
describe arcs intersecting in n , and join 
Bn ’ it will bisect the given angle. 

Proof, (th. 21, b. 1). 


B 









INSTRUMENTS, 


49 


PROBLEM III. 

From a given point, in a given line, to draw a perpendicu¬ 
lar to that line. 

Let AB be tlie given line, and C 
the given point. Take n and rn at equal 
distances on opposite sides of C; and 
from the points m and n, as centers, 
with any radius greater than nC or 
77i C, describe arcs cutting each other 
in S, Join BC, and it will be the per¬ 
pendicular required. 

Proof, (tli. 23, b. 1, cor.). 

The following is another method, which 
is preferable when the given point, C, is 
at or near the end of the line. 

Take any point, O, which is manifestly 
one side of the perpendicular, and join 
OC / and with OC, as a radius, describe 
an arc, euttkig AB in m and C. Join mO, and produce 
it to meet the arc, again, in n / mn is then a diameter to the 
circle. Join Cn, and it will be the perpendicular required. 

Proof, (th. 9, b. 3). 

PROBLEM IV. 

From a given point without a line, to draw a perpendicu 
lar to that line. 

Let AB be the given line, and C the 
given point. From C draw any oblique 
line, as Cn. Find the middle point of 
Cn by (Prob. I.), and from that point 
as a center describe a semicircle, having 

° A 

Cn as a diameter. From m, where the 
semi-circumference cuts AB, draw Cm, 
and it will be the perpendicular required. 

Proof, (th. 9, b. 3). 




s 









I 


50 SURVEYING AND NAVIGATION. 

PROBLEM V. 

At a given p>oint in a line, to make an angle equal to an¬ 
other given angle. 

Let A be tbe given point in the line 
AB, and DCE the given angle. c 

From C as a center, with any radius 
CE, draw the arc EE. 

From A as a center, with the radius A 
AF=CE\ describe an indefinite arc; and 
from F as a center, with EG as a radius, equal to ED , de* 
scribe an arc, cutting the other arc in G, and join AG / 
GAF will be the angle required. 

Proof, (th. 5, b. 3). 

PROBLEM VI. 

From a given point, to draw a line parallel to a given line. 

Let A be the given point, and CB the 
given line. Draw AB, making an angle, 

ABC / and from the given point A, in the 
line AB, draw the angle BAD—ABC, by 
the last problem. 

AD and CB make the same angle with AB / they are, 
therefore, parallel. (Th. 7, b. 1, cor. 1). 

PROBLEM VII. 

To divide a given line into any number of equal parts. 

' V .. . 

Let AB represent the given line, d 

and let it be required to divide it into 
any number of equal parts, say five. 

From one end of the line A, draw 
AD, indefinite in both length and 
position. Take any convenient dis- A * 

tance in the dividers, as Aa, and set it off on the line AD , 
thus making the parts Aa, ah, bo, &c., equal. Through the 











INSTRUMENTS. 

v V 


51 


last point, e, draw EB, and through the points a, b, <?, and d, 
draw parallels to &Z? (problem 6); these parallels will divide 
the line as required. 

Proof (th. 17, b. 2). 

PROBLEM VIII. 

To find a third proportional to two given lines. 


Let AB and .A (7 be any two lines. Place a_b 

them at any angle, and join CB. On the a _c 

greater line, AB , take AD = AC, and 2 

through D draw DE parallel to BC / AE ^ / \ 
is the third proportional required. \ \ 

Proof, (th. 17, b. 2.) \ \ 

a d b 

PROBLEM IX, 


To find a fourth proportional to three given lines. 


Let AB, AC', AD, represent the 
three given lines. Place the first two 
together, at a point forming any angle, 
as BA C, and join B C. On AB place 
AD, and from the point D, draw (pro¬ 
blem 6) DE parallel to BC / AE will 
be the fourth proportional required. 

Proof, (th. 17, b. 2). 


a_B 

a _o 

A_D 

C 



PROBLEM X. 


To find the middle, or mean proportional, between two 
given lines. 


Place AB and BC in one right line, 
and on as a diameter describe a 

semi-circle (postulate 3), and from the 
point B draw BD at right angles to 
A C (problem 3) ; BD is the meau pro¬ 
portional required. 

Proof, (cor. to th. 17, b. 3). 


A . 
B- 


B 


C 


D 



B 














52 


SURVEYING AND NAVIGATION. 


PROBLEM XI. 

To find the center of a given circle. 

Draw any two chords in the given cir¬ 
cle, as AB and CD q and from the middle 
point, n, of AB , draw a perpendicular to 
AB ; and from the middle point, m , draw c 
a perpendicular to CD • and where these 
two perpendiculars intersect wil] he the 
center of the circle. 

Proof, (th. 1, b. 3, cor). 

PROBLEM XII. 

» 

To draw a tangent to a given circle, from a given point , 
either in or without the circumference of the circle. 

When the given point is in the circum¬ 
ference, as A, draw AC the radius, and 
from the point A, draw AB perpendicular 
to ACj AB is the tangent required. 

Proof, (th. 4, b. 3). 

■« #• 

When A is without the circle, draw 
AC to the center of the circle; and on 
AC, as a diameter, describe a semi-cir¬ 
cle ; and from B , where the semi-cir¬ 
cumference cuts the given circumfer¬ 
ence, draw AB , and it will be tangent 
to the circle. 

Proof, (th. 9, b. 3), and (th. 4, b. 3). 

PROBLEM XIII. 

On a given line , to describe a segment of a circle , that 
shall contain an angle equal to a given angle. 



A B 











INSTRUMENTS. 


53 


Let AB be the given line, and C 
the given angle. At the ends of the 
given line, make angles DAB , DBA , 
each equal to the given angle, C. Then 
draw AE, BE\ perpendiculars to AD , 

BD • and from the center A, with 
radius EA or EB , describe a circle; then AFB will be the 
segment required, as any angle F , made in it, will be equal to 
the given angle C. 

Proof, (tli. 11, b. 3), and (tli. 8, b. 3). 

PROBLEM XIV. 

To cut a segment from any given circle , that shall contain 
a given angle. 

Let C be the given angle. Take 
any point, as A , in the circumference, 
and from that point draw the tangent 
AB / and from the point A , in the 
line AB , make the angle BAD — C 
(problem 5), and AED is the segment 
required. 

Proof, (tli. 11, b. 3), and (tli. 8. b. 3). 

PROBLEM XV. 

To construct an equilateral triangle on a given finite straight 
line. 

Let AB be the given line, and from one 
extremity, A, as a center, with a radius 
equal to AB , describe an arc. From the 
other extremity, B , with the same radius, 
describe another arc. From where 
these two arcs intersect, draw CA and CBj 
ABC will be the triangle required. 

Ihc, construction is a sujficient demonstration. Or (ax. 1). 


c 




D 








54 


SURVEYING AND NAVIGATION. 


E 

C 


A 


F 


D 



PROBLEM XVI. 

To construct a triangle having its three sides equal to three 

given lines , any two of which shall be greater than the third. 

* 

Let AB , CD) and EF represent tlie three 
lines. Take any one of them, as AB , to be 
one side of the triangle. From A as the 
center, with a radius equal to EF , describe an 
arc; and from B as a center, with a radius 
equal to CD , describe another arc, cutting 
the former in n. Join An and Bn , and 
AnB will be the A required. 

Proof, (ax. 1). 

PROBLEM XVII. 

To describe a square on a given line . 

Let AB be the given line, and from the q q 

extremities, A and B , draw AC and BD 
perpendicular to AB. (Problem 3.) 

From A , as a center, with AB as radius, 
strike an arc across the perpendicular at Cq 
and from (7, draw CD parallel to AB q 
ACDB is the square required. 

Proof, (tli. 26, b. 1). 


B 


PROBLEM XVIII. 

To construct a rectangle , or a parallelogram , whose adjacent 
sides are equal to two given lines. 

Let AB and AC be the two given lines, a _c 

From the extremities of one line, draw per- a__b 

pendiculars to that line, as in the last problem; and from 
these perpendiculars, cut off portions equal to the other line; 
and by a parallel, complete the figure. 

When the figure is to be a parallelogram, with oblique 
angles, describe the angles by problem 5. 

Proof, (tli. 26, b. 1). 









V 


INSTRU MENTS. 


55 


PROBLEM XIX. 

T-> describe a rectangle that shall be equivalent to a given 
square and have a side equal to a given line. 

Let AB be a side of the given square, and c_ d 

CD one side of the required rectangle. a_b 

Find the third proportional, EF , to CD e_f 

and AB (problem 8). Then we shall have, 

CD : AB : : AB : EF 

Construct a rectangle with the two given lines, CD and 
EF (problem 18), and it will be equivalent to the given 
square, (tin 3, b. 2). 

PROBLEM XX. 

To construct a square that shall be equivalent to the differ¬ 
ence of two given squares. 

Let A represent a side of the greater of two given squares, 
and B a side of the lesser square. 

On A , as a diameter, describe a semi¬ 
circle, and from one extremity, p, as a 
center, with a radius equal to B , describe 
an arc, n, and, from the point where it 
cuts the circumference, draw run and npj 
mn is the side of a square, which, when 
constructed, (problem 17), will be equivalent to the difference 
of the two given squares. 

Proof, (th. 9, b. 3, and 36, b. 1). 

PROBLEM XXI. 

To construct a square , that shall be to a given square , as a 
line M to a line N. 

Place M and iV r in a line, and 
on the sum describe a semicircle. 

From the point where they meet, 
draw a perpendicular to meet the 
circumference in A. Join Am and 
An , and produce them indefinitely. 
















56 


SURVEYING AND NAVIGATION. 


On An or An produced, take AC = to the side of the given 
square; and from C draw BC parallel to mn • AB is a side 
of the required square. 

PROBLEM XXII. 

The angles and one side of a triangle being given, to find 
ly construction the other sides. 

Draw the given side. From 
the ends of it lay off the angles 
that are adjacent to the given 
side; extend the other sides un¬ 
til they intersect. The dis¬ 
tances from the point of inter- 

r . . , A D B 
section to the extremities of the 

given line applied to the same scale of equal parts, from which 
the given line was taken, will give the other sides. 

% 

Example. 

One side of a triangle is 24 chains, and the adjacent 
angles are 37° and 64°. Required the other sides. 

Take from the scale of equal parts 24 chains, calling 10 
chains an inch, and draw the line AB. At A construct an 
angle of 37°, and at B an angle of 64°. Draw the lines meet¬ 
ing at C. Then AC applied to the scale will give 22, and 
BC applied will give 14.7 chains. If the altitude of the tri¬ 
angle is wanted, draw from C a perpendicular to AB, by 
problem 4. Then apply CD to the scale, and it will give 13.2 
chains for the altitude. 

PROBLEM XXIII. 

Two sides and an opposite angle of a triangle being 
given, to find the remaining side and the other angles by 
construction. 

Draw one of the given sides; from one end of it, lay off 
the given angle ; and extend a line indefinitely, from which 


c 





INSTRUMENTS. 


57 


the required side is to he 
taken. From the other end 
of the first side, with the 
remaining given side for 
radius, describe an arc cut¬ 
ting the indefinite line. 

The points of intersection 
will determine the required 
triangle. If the radius is 
such that the arc touches 
the indefinite line, as at D , 
the triangle will be right angled. If the arc does not intersect, 
the problem is impossible. 

Example . 

Two sides of a triangle are 21 and 25 chains and the angle 
opposite 21 is 16°. Required the third side. 

From the scale of equal parts, calling 10 chains an inch, 
take 25 chains for AB , the base of the triangle. At A con¬ 
struct an angle of 46°, and draw the indefinite line A C. 

Then take from the same scale of equal parts 21 chains, and 
with one foot of the dividers at B , and with 21 as radius, de¬ 
scribe an arc cutting the indefinite line in C and O' ; and AC 
or AO will be the required side. These applied to the scale 
will give A 0= 28.2 chains, or AO—QA chains. This exam¬ 
ple admits of two answers. 

PROBLEM XXIV. 

Given two sides and the included angle to construct the tri 
angle , and to measure the third side . 

Let the given sides be 23 and 28 
chains, and the included angle 51°. 

Take 28 from the scale of equal 
parts, and draw the line AB equal 
that length. At A construct an 
angle of 51°, with the line of 


c 




A 


B 




58 SURVEYING AND NAVIGATION. 

chords, or with the protractor; draw the line A C, and make it 
23 from the scale, and join BC; then apply BO to the same 
scale, and it will be found equal to 22.2 chains. 

The angles can also be measured by applying the protractor 
with the center at the angle to be measured. 

Or, from the line of chords take 60 as radius, and describe 
a circle to intersect the sides that include the angle to be mea¬ 
sured, the centre being at the angle. The distance between 
the points of intersection, applied to the line ot chords, will 
give the angle in degrees. 


PROBLEM XXV. 

To measure a 'parallelogram with a scale of equal parts. 

Let A BCD be a paral¬ 
lelogram. Take the base 
AB , and with the di¬ 
viders apply it to a scale 
of equal parts, and this 
will give the relative / 
length of AB. A 

Then from one of the angles as at D , draw a perpendicular 
to AB by problem 4, as DE; then apply DE to the same 
scale of equal parts, and this will give the length of DE rela¬ 
tively. If ABCD is constructed on a scale of 10 chains to the 

« 

inch, then will AB = 25 chains, and DE — 13.2 chains. 

These are sufficient to determine the area of the parallelo¬ 
gram. 



PROBLEM XXVI. 

To measure a trapezium. 

Let ABCD be any trapezium; draw the diagonal AC. 
From D draw DE parallel to A C, and join EC j then the 
triangle EBC will be equivalent to the trapezium ABCD. 
Since DE is parallel to AC, the triangle ADC is equivalent 





INSTRUMENTS. 


i 


59 


to AEG ; to each add the triangle ABO \ then will result 
ABGD-EBO. 

Now apply EB to 
a scale of equal parts, 
and draw a perpendi¬ 
cular from C to AB , 
and apply that to the 
same scale. 

If ABCE> is drawn 
to a scale of 10 chains to the inch, then will AB — 20 
chains, EB = 34.5 chains, and the altitude of the triangle 
EBC will be 13.2 chains. These lines will determine the 
measure of the triangle, and consequently of the trapezium. 

PROBLEM XXVII. 

To measure instrumentally a figure of five sides . 

Let ABODE be any 
figure of five sides; draw 
the diagonals A D and 
BD y then draw EG par¬ 
allel to AD , and OF par¬ 
allel to BD * then join 
GD and FD y then will 
the triangle GFD be equi¬ 
valent to the figure ABODE. Since EG is parallel to AD. 
the triangle AED—AGD / and since OF is parallel to DB, 
we have BCD—BED y whence it follows that 

GFD = ABODE. 

Then apply GE to the scale of equal parts, and if AB — 
20 chains, then will GF — 32 chains, and the perpendicular 
from D on AB will be 17.4 chains, which is the altitude of 
the triangle. 

These lines determine the measure of the figure ABODE. 





C 







60 


SURVEYING AND NAVIGATION. 



PROBLEM XXVIII. 

To measure any figure of six sides. 

Draw the diagonals 
EA and DB ; draw 
FI1 parallel to EA, and 
CG parallel to DB; 
then join Eli and DG. % 

Draw the diagonal Dll , 

and draw if/ parallel to I HA B G 

DH and join Dfi and IGD will be equivalent to ABCDEF. 
Since FII 'is parallel to AE^ we have AEE=AIIE; and since 
CG is parallel to DB , we have BCD—BGD • and also since 
El is parallel to Dll , we have HED — I1ID • whence it fol¬ 
lows that IGD — ABCDEF. 

Now, if we apply IG to a scale of equal parts, we shall get 
its length, and we find that if AB is 12 chains, then IG = 
33.5 chains; we also find that a perpendicular from D to the 
base AB will equal 17.3 chains. 

Wherefore the measure of ABCDEF is determined. 













I 



CHAPTER II. 

LOGARITHMS AND TRIGONOMETRY. 


Mensuration, Surveying and Navigation are practical appli- 
cations of geometry and trigonometry. 

The geometrical principles involved can be best acquired by 
the study of that science in its full extent, as laid down in the 
common elementary text-books. 

The principles of trigonometry required do not, however, 
include all those given in the text books upon that subject, or 
all that are important to the general student. 

It is therefore considered best to give a brief resume of those 
theorems practically involved in the subsequent matter of the 
work, for the purpose of refreshing the recollection of those 
who have already pursued the study, and as a condensed course 
for those desiring to fit themselves immediately for practice. 

It is intended to include in this chapter only those matters 
intimately connected with practice. For a fuller discussion of 
both Logarithms and Trigonometry, the student is referred to 
Robinson’s New Geometry and Trigonometry. 

For greater ease in the solution of almost all the practical 
problems of Mensuration, Surveying and Navigation, a clear 
knowledge of Logarithms is essential; and the student should 
make himself perfectly familiar with their use. 

* The student who is acquainted with Logarithms and Trigonometry, and 
desires to take up Surveying, may omit this entire chapter if he chooses, 
and also so much of Mensuration in the following chapter as may be deemed 
advisable. 


62 


SURVEYING AND NAVIGATION. 


SECTION I. 

OF LOGARITHMS. 

The Logarithm of any quantity is an exponent expressing 
the power to which a constant or fixed quantity must be raised 
to equal the given quantity. Thus, in the equations 

a n = x , 
a m = y , 

where a is a constant, n is the logarithm of x, and m is the 
logarithm of y. 

Likewise in the numeral equations, 

10 2 = 100 , 

10 3 = 1 , 000 , 

where 10 is the constant, 2 is the logarithm of 100, and 3 is the 
logarithm of 1,000. 

Thus logarithms are signs , showing the relation which dif¬ 
ferent quantities bear to one assumed or constant quantity. 

In many operations it is found far more convenient to com¬ 
pare quantities by their relations than absolutely, and logarithms 
have thus become of vast utility to the practical mathe¬ 
matician. 

A System of Logarithms comprises the logarithms of nurn 
bers derived from any assumed constant quantity. 

This assumed constant is called the base of the system, and 
any number being taken as the base, a system of logarithms 
may be formed from it. 

The main qualification is that of convenience. Two systems 
have been prepared, called from the projectors, Napier’s and 
Briggs’ logarithms. The one in common use is the Briggs’ 
system, which has for its base the number 10. 

In writing the logarithm of a number or letter, the contrac¬ 
tion “log.”-is used. Thus, in the equation a n =x , we have 

n = log. x. 


LOGARITHMS. 


63 


In tlie Briggs system, where 10 is the base, it follows from 
the definition of a logarithm, that as 10 5 =100,000, 5=log. 
100,000. In like manner w T e have 

10 4 = 10,000, whence 1 = log. 10,000; 


10 3 

— 

1,000, 

u 

3 = log. 1,000 

10 2 

— 

10,0, 

u ■ 

2 = log. 100; 

10 1 

— 

10, 

a 

1 = log. 10 ; 

10° 

— 

1, 

u 

0 = log. 1; 

10- 1 

— 

•1, 

u 

— 1 = log. .1 ; 

10 — 2 

= 

.01, 

u 

—2 = log. .01; 

o 

1 

w 

= 

.001, 

u 

— 3 = log. .001; 


From an examination of the above numbers and their loo-- 
arithms, it is seen that as the logarithm of 1 is 0, and the log¬ 
arithm of 10 is 1,—the log. of any number between 1 and 10 
will be greater than 0, and less than 1; that is, it will be a 
decimal. So also the logarithm of any number between 10 
and 100 will be greater than 1 and less than 2 : that is, it will 
be 1 and a decimal. lieturning to 0 = log. 1, we find that —1 
is the log. of .1. Now any number between 1 and .1—that is, 
any decimal greater than .1—will have a logarithm greater 
than —1, but still negative or less than 0 ; that is, the loga¬ 
rithm will be —1 plus a decimal. So taking —l = log. .1, and 
—2 = log. .01, any number between .1 and .01 will have for its 
logarithm a number less than —1, and greater than —2; or, 
—2 plus a decimal. 

It will also be noticed that while the quantities increase and 
decrease by multiplying or dividing by 10, the logarithms in* 
crease and decrease by the addition or subtraction of 1, which 
accords with the algebraic rule, that adding exponents multi¬ 
plies, and subtracting exponents divides the quantities them¬ 
selves. For example, 

10 1 = 10 , 

10 2 = 100 . 

10 1+2 = 10 3 =s 1,000. 


Multiplying, 


64 


SURVEYING AND NAVIGATION. 


Tims increasing or decreasing any number in a tenfold ratio, 
will increase or decrease the logarithm by unity. 

From the foregoing are deduced the following: 

1. All powers of 10 and their reciprocals will have for log¬ 
arithms integral numbers. 

2. All other numbers will have for logarithms an integer 
together with a decimal, or simply a decimal. 

3. Increasing or decreasing numbers in a tenfold ratio, in¬ 
creases or decreases only the integer, without affecting the 
decimal of the logarithm. 

4. The integer of the logarithm depends entirely upon the 
local value of the number, or the position which the figures 
occupy with reference to the decimal point. 

5. The integer of the logarithm* of any number greater than 
unity will always be one less than the number of integral 
places which that number contains. 

6 . The integer of any number less than unity will always 
be equal to the number of places the first significant figure is 
removed from the decimal point. 

7. The integer belonging to the logarithm of a decimal, 
will always have the minus sign. 

8. The decimal, however connected with this negative in* 
teger, will he positive. 

From the integer of a logarithm, the position of the first 
significant figure with reference to unity may always be 
determined. 

Remark.— To make clear the distinction between a logarithm to be sub 
traded , and one simply having a negative integer, the minus sign in the 
latter case is placed above the integer, not before it, as in the other case. 

The integral portion of a logarithm is called its character - 
istic or index . 


LOGARITHMS. 


65 


LOGARITHMIC TABLE. 

It is now necessary to explain the table of logarithms and 
its nse. 

In the table connected with this work (as is the usual form), 
the logarithms of numbers from 1 to 100 are given entire, with 
both index and decimal. Thus the logarithm of any number 
within those limits may be taken at once complete from the 
table. 

For numbers above 100 to 10,000, only the decimals are 
given; for, as the index depends entirely upon the position of 
the highest significant figure, it may be readily supplied. Thus, 
the decimal of the logarithm of 7956, as found in the table, 
is .900695 ; and since the number has four integral places, we 
have only to prefix the integer 3, and we have 

3.900695 = log. 7956. 

Should we divide the number successively by 10, we must 
subtract 1 each time from the logarithm ; hence 

2.900695 = log. 795.6, 

1.900695 = “ 79.56, 

0.900695 = “ 7.956. 

In the table, the decimal part of the logarithms of all these 
numbers is the same, the significant figures alone being con¬ 
sidered. The index gives the position of the figures with 
reference to unity. 

i 

1. To find the logarith m of any number f rom the table. 

The logarithm of a number containing three figures will be 
found with great ease. 

The figures are arranged in a column, headed N., at the left 
of the page; in the second column, headed 0, opposite the 
proper figures in the first column, will be found the decimal 
of the logarithm. 


66 


SURVEYING AND NAVIGATION. 



If the number contain four figures, the first three will be 
found as before, the fourth figure will be found at the head of 
one of the columns, commencing with 0 at the left, and pass¬ 
ing to 9 at the right. 

In the column headed by the fourth figure, and opposite the 
first three in the left hand column, will be found the decimal 
of the logarithm. 

It is to be noted that, as the first two terms of the decimal 
are the same for several successive numbers, they are only 
given in the column at the left, headed 0, and must be prefixed 
to the four figures given in the other columns. 

Also, when dots are found in place of figures, the two lead¬ 
ing decimals must be taken from the line below, and ciphers 
used in place of the dots. For illustration take the following 
examples* 

1. To find the logarithm of the number 154. 

% - 

Turning to page 4 of the table of logarithms, in the column 
at the left headed N., we find the number 154; opposite, in 
the column headed 0, we find 7521, the last four figures of 
the decimal. Two lines above, we find 18 the first two figures 
to be prefixed. We thus have for the decimal of the loga¬ 
rithm, .187521. Since the number has three integral places, 
the index will be 2. Hence, 

Log. 154 = 2.187521, Ana. 

2. To find the logarithm of the number 3725. 

We find 372 at the side of the table, and in the column 
marked 5 at the top, and opposite 372, we find .571126, for the 
decimal part of the logarithm. Hence, 

Log. 3725 = 3.571126, Ans. 

3. To find the logarithm of the number 834785. 

This number is so large that we cannot find it in the table, 
but we can find the numbers 8347 and 8348. The logarithms 


LOGARITHMS. 


or 


of these numbers are the same as the logarithms of the num¬ 
bers 834700 and 834-800, except the indices. 

834700, log. 5.921530 
834800, log. 5.921582 

Differences, 100 52 

The given number is between two assumed numbers, and 
its logarithm must lie between the logarithms of those assumed 
numbers. Now the differences between the logarithms are 
very nearly proportional to the differences between the num¬ 
bers ; so nearly that, where the numbers are not too widely 
separated, for all practical purposes the proportion may be 
considered exact. AYe may, therefore, form the proportion, 

Difference between assumed numbers : 

Difference between lesser assumed and the given number:: 

Difference between log.’s of assumed numbers: 

Difference between log.’s of lesser and the given number. 

Using the differences found above, 


Or, 


100: 85:: 52:44.2, 
1: .85 : : 52 : 44.2. 


In the last proportion, the difference between the lesser 
assumed and the given number, considered as a decimal, and 
multiplied by the difference between the logarithms of the 
two assumed numbers gives the difference between the loga¬ 
rithm of that lesser number and the logarithm of the given 
number. This difference added to the logarithm of the lesser 
number, as a correction, gives the required logarithm. 


Thus, Log. 834700 = 5.921530 

Difference or correction, = 44.2 

Log. 834785 = 5.921574.2. 

The difference between the logarithms of the two assumed 
numbers, is called the Tabular Difference; and for conven¬ 
ience, this is given in the column headed 4, and in a line by 
itself. For example, on page 4, under column 4, in the sixth 





68 


SURVEYING AND NAVIGATION. 


line, we find 281, which is the Tabular Difference for the 
logarithms immediately preceding and following. 

From these illustrations we derive, for finding from the table 
the logarithm of a number consisting of more than four places 
of figures, the following 

RULE. 

Take from the table the logarithm of the number expressed 
by the four superior figures / this, with the proper index, is 
the approximate logarithm. Multiply the number expressed 
by the remaining figures of the number, regarded as a decimal, 
by the tabular difference, and the product will be the correction 
to be added to the approximate logarithm to obtain the true 
logarithm. 

EXAMPLES. 

1. What is the log. of 357.32514 ? 

The log. of 357.3 is. 2.553033 

No. not included, .2514 
Tabular difference, 122 

Frod., 30.6708 ; correction, 31 
log. sought, 2.553064 


The log. of 35732.514 is. 4.553064 

“ .035732514 “.— 2.553064. 

2. What is the log. of 7912532 ? 

Approximate log.,. 6.898286 

.532 x 55 = correction,. 29 


True log. = 6.898315, A ns. 

2. A logarithm being given, to find its corresponding 
number. 

For example, what number corresponds to the log. 
6.898315? 










LOGARITHMS. 


69 


The index 6 shows that the entire part ff the number must 
contain seven places of figures. With the decimal part, 
.898315, of the log., we turn to the table, and find the next 
less decimal part to be .898286, which corresponds to the 
superior places, 7912. 

The difference between the given log. and the one next less 
is 29. This we divide by the tabular difference, 55, be¬ 
cause we are working the converse of the preceding problem 
Thus, 

29-^55 = .52727+. 

Place the quotient to the right of the four figures before 
found, and we shall have 7912527.27 for the number sought. 

This example w T as taken from the preceding case, and the 
number found should have been 7912532; and so it would 
have been, had we used the true difference, 29.26, in place 
of 29. 

When the numbers are large, as in this example, the result 
is liable to a small error, to avoid which the logarithms should 
contain a great number of decimal places; but the logarithms 
in our table contain a sufficient number of decimal places for 
most practical purposes. 

Hence, for finding the number corresponding to any given 
logarithm, we have the following 

RULE. 

r 

Look in the table for the decimal part of the given loga¬ 
rithm , and if not found take the decimal next less , and take 
out the four corresponding figures. 

Take the difference between the given logarithm and the next 
less in the table; divide that difference by the tabular difference , 
and write the quotient on the right of the four superior figures,, 
and the result is the number sought. 

Point off the whole number required by the given index . 


70 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

1. Given tlie logarithm 3.743210, to find its corresponding 
number true to three places of decimals. Ans. 5536.177. 

2. Given the logarithm 2.633356, to find its corresponding 

number true to. two places of decimals. Ans. 429.89. 

3. Given the logarithm —3.291746, to find its correspond¬ 
ing number. A?is. .0019577. 

4. What number corresponds to the log. 3.233568 ? 

Ans. 1712.25. 

5. What is the number of which 1.532708 is the log. ? 

- Ans. 34.0963. 

6. Find the number whose log. is 1.067889. 

Ans. 11.692. 

T~ ■ • 

APPLICATIONS OF LOGARITHMS. 

From the definitions and principles heretofore given, the 
rules for applying logarithms may be readily deduced. 

'»- r. 1 

To multiply by logarithms. 

RULE. 

Add the logarithms of multiplicand and multiplier; the 
sum will be the logarithm of the product. From the table 
■find the number corresponding to this logarithm 3 and it will 
be the product repaired. 

EXAMPLES. 

1. What is the product of 7896 and 9872 ? 

Log. 7896 = 3.897407 
“ 9872 = 3.994405 

Log. of product = 7.891812 
Approximate log. = .891760 = log. 7794 

* V - 

Difference = 52 

Tabular difference = 56. 

Correction j§ = .928571. 

Therefore, 7.891812 = log. 77949285.71. 

Ans. 77949285.71 nearly. 

•» 




LOGARITHMS. 


71 


Remark. —As the logarithms are not exact, the correction carried out 
beyond two places becomes inaccurate;—hence the result in the preceding 
example is incorrect, though the error is slight. When great accuracy is 
required, and large numbers are used, the logarithms must be more accu¬ 
rately calculated, and carried out to more decimal places, or their use dis¬ 
pensed with. 

2. Required the product of 976.24 and 9.76. 

Ans. 9528.11. 

3. Required the continued product of 8.761, 3.426, 7.97, 

and 5.63. Ans. 1346.814+ . 

4. Required the continued product of 9.913, 5.864, 11.23, 
4.51, and 7.62. 


To divide by logarithms. 

As before, logarithms are considered as exponents. By 
algebraic rule for division, the difference between exponents 
of dividend and divisor will be the exponent of the quotient. 
Hence the following 

RULE. 

Subtract the logarithm of the divisor , from that of the 
dividend / the result will be the logarithm of the quotient. 
Find the number corresponding to this from the table , and it 
vnll be the quotient required. 

EXAMPLES. 

1. Divide 8967.42 by 32.1. 

Log. 8967.42 = 3.952668 
« 32.1 = 1.506505 

Log. of quotient = 2.446163 
Approximate log. = 2.446071 = log. 2793 

Difference == 92 

Tabular difference == 155 

Correction = 92 ~ 155 = .59 
Hence, 2.446163 = log. 279.359 


Ans. 279.359, 




72 SURVEYING AND NAVIGATION. 

2. What is the quotient of 739.86 divided by 23.12? 

Ans. 

3. What is the value of the fraction r \ 7 T 6 T expressed 

decimally ? Ans. 

4. What is the quotient of 36278 divided by 97 ? 

To solve proportions by logarithms. 

RULE. 

Add the logarithms of the means, and subtract that of the 
given extreme / or add the logarithms of the extremes, and 
subtract that of the given mean. The result in either case will 
be the logarithm of the required term, which will be found by 
taking from the table the number corresponding to the loga¬ 
rithmic result. 

The reason for this rule is evident from the rule for solving 
proportions by multiplication and division, and the rules 
before given for the use of logarithms. 

EXAMPLES. 

1. Required the fourth term in the proportion 

97 : 126 = 321 : ? 

Log. 126 = 2.100371 
Log. 321 = 2.506505 

4.606876 

Subtract log. 97 = 1.986772 

Log. extreme = 2.620104 

2.620104 = log. 416.97. Ans. 

2. Required the fourth term in the proportion 

32.71 : 142.81 ±= 76.4 : ? Ans. 333.56. 

3. Required the third term in the proportion 

43.24 : 217.16 = ? : 137.39. Ans. 27.35. 


/ 


I 




LOGARITHMS. 


73 


In tlie solution of proportions by logarithms, and in other 
eases where division is performed, the Arithmetical Comple¬ 
ment of the logarithm of the divisor is often used. 

The Arithmetical Complement of any number is the re¬ 
mainder after subtracting that number from a unit of the next 
higher order. 

To obtain the complement, subtract each figure, commencing 
at the left , from 9, save that upon the right , which subtract 
from 10. 

To solve a proportion with the use of the arithmetical com¬ 
plement, we have the following 

RULE. 

First obtain the complement of the logarithm of the term to 
be used as a divisor ; to this add the logarithms of the two 
terms to be multiplied , subtract 10 from the result , and there 
will remain the logarithm of the term required. 

To apply to simple division this method, add the comple¬ 
ment of the logarithm of the divisor to the logarithm of the 
dividend, and diminish the result by 10, or by 100 should the 
logarithm of the divisor exceed 10. 

That the use of the complement does not change the result 
is evident from the following equation. 

((10— a ) -f- b T c ^ —10 b -)- c — a , 
where a— log. of divisor, b and <?=log.’s of factors of the 
dividend. 

EXAMPLES. 

1. Find the unknown term in the proportion 

376 : x = 497 :1891 

Co. log. 497 = 7.303644 
log. 376 =' 2.575188 
log. 1891 = 3.276692 

Bum less 10 = log. x = 3.155524 

4 



I 


74 SURVEYING AND NAVIGATION. 

Sum less 10 = log. x = 3.155524 
App. log.=log. 1430 — .155336 

Difference, = 188 

Tab. Diff. ==304. 188-^304=62 correction. 

Hence, 3.155524=log. 1430.62 Ans. = 1430.62. 

2. Required the unknown term in the proportion 

3796 : 9843 =4265 : x. 

3. Required the unknown term in the proportion 

472: 976=a?: 2345. 

4. Required the unknown term in the proportion 

7.693 :11 = 9.679 : x. 

To perform involution by logarithms . 

RULE. 

To involve a quantity to any power, multiply its logarithm 
by the index of the power; the product will be the logarithm of 
the power required. 

For if 10 2 =100, when 2 = log. 100, raising both sides to 
second power, 10 4 =100 2 = 10000. In this equation 4=2 x 
2=log. 10000 ; that is, the log. of 100, multiplied by the index 
2, gives the log. of the square of 100. 

EXAMPLES. 

1. Required the cube of 32. 

Log. 32 = 1.505150 
Multiply by 3 

Log. of power = 4.515450 

App. log. = 515344= log. 3276 

106-^-132 = 737 

4.515450=log. 32767.37 

Ans. by logarithms, 32767.37 
“ “ multiplication, 32768. 





LOGARITHMS. 


75 


The error of .63 in the first result is caused by the in accu¬ 
racy of the logarithms, which generally affects results beyond 
the sixth or seventh place from the left. 

2. Required the fourth power of 2.763. 

3. Required the ninth power of .0176. 

Note.-.—T he decimal of every logarithm is positive. 


To perform evolution by logarithms . 

RULE. 

Divide the logarithm of the quantity by the index denoting 
the root required ; the result will be the logarithm of the root 
desired. 

This rule would follow like the preceding from the algebraic 
rule for the same operation. 


EXAMPLES. 

1. Required the cube root of 7896.34. 

Log. of 7896.34 = 3.897426 
Divided by 3 = 1.299142 

1.299142 = log. 19.913, Ans. 

2. Required the fifth root of 9764. 

Ans. 6.279. 

3. Required the cube root of 89763. 

Ans. 44.774. 


4. Required the sixth root of 97643.89. 


76 


SURVEYING AND NAVIGATION. 


SECTION II. 

OF PLANE TRIGONOMETRY. 

Note. —References to geometry refer to Robinson’s New Geometry and 
Trigonometry; those to plane and spherical trigonometry refer to the sec¬ 
tions on those subjects which follow. B. A. and 0. A. Sj3her. Trig, are 
used to distinguish propositions, concerning right-angled and oblique- 
angled triangles. When the numbers alone of propositions or proportions 
are given, they refer to those in the same section. 

Trigonometry, in its literal and restricted sense, has for its 
object the measurement of triangles. When it treats of plane 
triangles it is called Plane Trigonometry. In a more enlarged 
sense, trigonometry is the science which investigates the rela¬ 
tions of all possible arcs of the circumference of a circle to 
certain straight lines, termed trigonometrical lines or circular 
functions , connected with and dependent on such arcs, and 
the relations of these trigonometrical lines to each otiier. 

The measure of an angle is the arc of a circle intercepted 
between the two lines which form the angle—the center ot 
the arc always being at the point where the two lines meet. 

The arc is measured by degrees , minutes , and seconds ; there 
being 360 degrees to the whole circle, 60 minutes in one 
degree, and 60 seconds in one minute. Degrees, minutes, and 
seconds are designated by °, ', "; thus, 27° IP, 21", is read 
27 degrees IT minutes 21 seconds. 

The circumferences of all circles contain the same number 
of degrees, but the greater the radius, the greater is the abso¬ 
lute length of a degree. The circumference of a carriage 
wheel, the circumference of the earth, or the still greater and 
indefinite circumference of the heavens, has the same number 
of degrees; yet the same number of degrees in each and every 
circumference is the measure of precisely the same angle. 


TRIGONOMETRY. 


77 


DEFINITIONS. 

1. The Complement of an arc is 90° minus the arc. 

2. The Supplement of an arc is 180° minus the arc. 

3. The Sine of an angle, or of an arc, is a line drawn from 
one end of an arc, perpendicular to a diameter drawn through 
the other end. Thus, BF\§ the sine of the arc AB , and also 
of the arc BDE. BK is the sine of the arc BD. 

4. The Cosine of an arc is the perpen¬ 
dicular distance from the center of the 
circle to the sine of the arc; or, it is the 
same in magnitude as the sine of the com¬ 
plement of the arc. Thus, CF is the 
cosine of the arc AB ; hut CF = KB , is 
the sine of BD. 

5. The Tangent of an arc is a line touching the circle in 
one extremity of the arc, and continued from thence, to meet 
a line drawn through the center and the other extremity. 
Thus, All is the tangent to the arc AB , and DL is the tan¬ 
gent of the arc DB. 

6. The Cotangent of an arc is the tangent of the comple¬ 
ment of the arc. Thus, DL , which is the tangent of the arc 
DB, is the cotangent of the arc AB. 

Remark. —The co is but a contraction of the word complement. 

7. The Secant of an arc is a line drawn from the center of 
the circle to the extremity of the tangent. Thus, CII is the 
secant of the arc AB, or of its supplement BDE. 

8. The Cosecant of an arc is the secant of the complement. 
Thus, CL, the secant of BD, is the cosecant of AB. 

9. The Versed Sine of an arc is the distance from the 
extremity of the arc to the foot of the sine. Thus, AF is 
the versed sine of the arc AB, and DIC is the versed sine of 
the arc DB. 

For the sake of brevity, these technical terms are con 
tracted thus: for sine AB, we write sin. ABj for cosine 










78 


SURVEYING AND NAVIGATION. 


AB , we write cos. AB j for tangent AB , we write tan. 
AB , etc. 

From the preceding definitions we deduce the following 
obvious consequences: 

1st. That when the arc AB becomes insensibly small, or 
zero, its sine, tangent, and versed sine are also nothing, and 
its secant and cosine are each equal to radius. 

2d. The sine and versed sine of a quadrant are each equal 
to the radius; its cosine is zero, and its secant and tangent 
are infinite. 

3d. The chord of an arc is twice the sine of one half the 
arc. Thus, the chord BG is double the sine BF. 

4th. The versed sine is equal to the difference between the 
radius and the cosine. 

5th. The sine and cosine of any arc form the two sides of 
a right-angled triangle, which has a radius for its hypotenuse. 
Thus, GF and FB are the two sides of the right-angled tri¬ 
angle CFB. 

Also, the radius and tangent always form the two sides of 
a right-angled triangle, which has the secant of the arc for 
its hypotenuse. This we observe from the right-angled tri¬ 
angle CAR. 

To express these relations analytically, we write 

sin. 2 + cos. 2 = R 2 (1) 

B 2 + tan. 2 = sec. 2 (2) 

From the two equiangular triangles CFB , CAR , we have 

CF: FB = CA: AR. 

That is, 

T) • 

cos. : sin. = R : tan.; whence, tan. = l sm ‘ . (3) 

cos. 

Aiso, 

CF: CB = CA : CR. 

That is, 

cos.: R — R : sec.; whence, cos. sec. = R 2 . (4V 









TRIGONOMETRY. 


79 


The two equiangular triangles, CAII and CDL, give 

CA : All = DL : 7>6 y . 

That is, 

i? : tan. = cot. : R • whence, tan. cot. = R 2 . (5) 

Also, 

CF: FB — DL : DC. 

That is, 

cos. : sin. = cot. : B ; whence, cos. R — sin. cot. (6) 

From equations (4) and (5), we have 

cos. sec. = tan. cot. (7) 

Or, 

cos. : tan. = cot. : sec. 

We also have 

ver. sin. = R — cos. (8) 

The ratios between the various trigonometrical lines are 
always the same for arcs of the same number of degrees, 
whatever be the length of the radius; and we may, therefore, 
assume radius of any length to suit our convenience. The 
preceding equations will be more concise, and more readily 
applied, by making the radius equal unity. This supposition 
being made, we have, for equations 1 to 6, inclusive, 

sin. 2 + cos. 2 = 1. 

1 + tan. 2 = sec. 2 


tan. = 55: (3) 

cos. 

tan. = -J— (5) 

cot. ' 

Let the circumference, 
AEDH , be divided into four 
equal parts by the diameters, 
AD and EH, the one hori¬ 
zontal and the other vertical. 
These equal parts are called 
quadrants , and they may be 
distinguished as the first , 
second, third, and fourth 
quadrants. 


cos. = 


cos. 


sec. 

sin. cot. 


a) 

( 2 ) 

(*) 


( 6 ) 


E 


















80 


SURVEYING AND NAVIGATION. 


The center of the circle is taken as the origin of dis¬ 
tances, or the zero point, and the different directions in which 
distances are estimated from this point are indicated by the 
signs + and —. If those from C to the right be marked -f, 
those from C to the left must be marked — ; and if distances 
from C upwards be considered plus, those from G downwards 
must be considered minus. 

If one extremity of a varying arc be constantly at A , and 
the other extremity fall successively in each of the several 
quadrants, we may readily determine, by the above rule, the 
algebraic signs of the sines and cosines of all arcs from 0° 
to 360°. Now, since all other trigonometrical lines can be 
expressed in terms of the sine and cosine, it follows that the 
algebraic signs of all the circular functions result from those 
of the sine and cosine. 

We shall thus find for arcs terminating in the 



Bin. 

cos. 

tan. 

cot. 

sec. 

coeec. 

vers. 

1st quadrant, 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

2d “ 

+ 

— 

— 

— 

— 

+ 

+ 

V# 

CO 

— 

— 

+ 

+ 

— 

— 

+ 

4th “ 

— 

+ 

— 

— 

+ 

— 

+ 


PROPOSITION I. 


The chord of 60° and the tangent of 45° are each equal to 
radius ; the sine of 30°, the versed sine of 60°, and the cosine 
of 60° are each equal to one half the radius. 


E 


H 


With C as a center, and CA as a radius, 
describe the arc ABF , and from A lay off 
the arcs vlZ)=45 0 , ^4i?=60 o , and ^4 jE= 90°; 
then is FB— 30°. 

1st. The side of a regular inscribed hexa¬ 
gon is the radius of the circle (Geom. Prob. 

28, B. IY), and as the arc subtended by 

each side of the hexagon contains 60°, we have the chord of 

of 60° equal to the radius. 









TRIGONOMETRY. 


81 


2d. The triangle CAII\s right-angled at A, and the angle 
C is equal to 45°, being measured by the arc AD ; hence 
the angle at II is also equal to 45°, and the triangle is isosceles. 
Therefore AII=CA — radius of the circle. 

3d. The triangle ABC is isosceles, and Bn is a perpendicu¬ 
lar from the vertex upon the base; hence An — nC = Bm. 
But Bm is the sine of the arc BE', Cn is the cosine of the arc 
AB, and An is the versed sine of the same arc, and each is 
equal to one half the radius. 

Hence the proposition; the chord of 60°, etc. 

PROPOSITION II. 

Given , the sine and the cosine of tioo arcs , to find the sine 
and the cosine of the sum and of the difference of the same 
arcs expressed by the sines and cosines of the separate arcs. 

Let G be the center of the circle, 

CD the greater arc, and DF the 
less, and denote these arcs by a and 
b respectively. 

Draw the radius GD / make the 
arc DE equal to the arc DF, and 
draw the chord EE. From F and 
E, the extremities, and 7, the mid¬ 
dle point of the chord, let fall the 
perpendiculars FM, EP , and IE, 
on the radius GC. Also draw DO, the sine of the arc CD, 
and let fall the perpendiculars IH on EM, and EK on IE. 

How, by the definition of sines and cosines, DO—§m.a; 
GO=uos.a; 77= sin7/ GI=cos.b. We are to find 

EM = sin. (a 4- b) ; GM = cos. {a -f b) ; 

EP = sin. {a — b) ; GP — cos. {a — b). 

Because IE is parallel to D 0, the two a’s, GDO, GIE, 
are equiangular and similar. Also, the A Fill is similar to 
the a GIE; for the angles, FIG and IIIE, are right angles; 


F 










82 


SURVEYING AND NAVIGATION. 


from these two equals, taking away the common angle HIG , 
we have the angle FIH— the angle GIN. The angles at H 
and N are right angles; therefore, the a’sj PHI, GIN , and 
GNO , are equiangular and similar ; and the side Klis homol¬ 
ogous to IN and NO. 

Again, as FI— IE\ and IK is parallel to FII, 

FII = IK, and III = KE. 

By similar triangles w r e have 

GD : DO = GI: IN. 

That is, II : sin.a = cos.J : IN; IN = cos ' h 


Also, GD: GO = FI: FH. 

That is, R : cos.® = sin. b : IIF/ or, FII = 

Also, GD : GO — GI: GN. 

That is, R : cos.® = cos .b : GN/ or GN = 

Also, GD : DO = FI: III. 

That is, R : sin.® = sin.Z> : III/ or, IK — 


R 

cos .a sin.6 
R 

cos.® cos. 5 
R 

sin .a sin.fr 

~R 


(i) 


( 2 ) 


(3) 




By adding the first and second of these equations, we have 
IN + FII = FII = sin. ( ®-f fr). 

rpi j • • / . 7 \ sin.® cos.fr + cos.® sin.5 

That is, sm. (®+fr) = - — -. 

R 

By subtracting the second from the first, since 

IN—FK — IN—IK ~ EP , we have 
sin.® cos.fr—cos.® sin.fr 


sin. (®—fr) = 


R 


By subtracting tlv> fourth from the third, we have 

GN—III = Gif = cos. (®+fr) for the first member. 

Hence, cos. (a+5) = cos ^ C06 - 8 - gin - a sin ^ 

By adding the third and fourth, we have 

GN+III — GN+NP = GP = cos. (a-b). 


(5) 









TRIGON OMETRY. 


83 


Hence, cos. (a-h) = c °s.& + sin.a sin.b 

B 


( 6 ) 


(^) 


Collecting these four expressions, and considering the radius 
unity, we have 

sin. (#+5)=sin.a cos.J-fcos. a sin .b (7) 

sin. (a —&)=sin.a cos.&—cos.a sin. b (8) 

cos. (a + b)=t‘os.a cos .b —sin.& sin .b (9) 

„ cos. (a — b)— cos.& eos.&-fsin.« sin.& (10) 

Formulae (A) accomplish the objects of the proposition, and 
from these equations many useful and important deductions 
can be made. The following are the most essential: 

By adding (7) to (8), we have (11); subtracting (8) from (7) 
gives (12). Also, (9) added to (10) gives (13); (9) taken from 
(10) gives (14). 

sin. (#+&) +sin. (a — b)= 2 m\.a cos.b (11) 

sin. (a-\- b) —sin. (a—b)=2 cos .a sin .b (12) 

cos. (« + J) + cos. (a—b )=2 cos.« cos.5 (13) 

„ cos. (a—b)— cos. (a+b) = 2 sin. a sin .b (14) 

If we put a + b=A , and a—b—B , then (11) becomes (15), 
(12) becomes (16), (13) becomes (17), and (14) becomes (18). 


(£)! 


(O) 


l 

sin.A + sin.Z?=2 sin.I 


/A + B\ np | 

< A -B\ 

4 2/ 

\ 2 / 

(A + B\ e . n j 

fA-B\ 

4 2/ 

i 2 / 

(A + B\ ^ a j 

<A-B\ 

\ 2 J ' ' 

l 2 / 

IA+B\ . / 


\ 2 / ' 

* 2 / 

observing that Sin ‘ = 

cos. 


(15) 

( 10 ) 

(U) 

(18) 


^Lr=cot.= —— } as we learn by equations (6) and (5), we 


sin. 

shall have 


tan. 


f 



















84 


SURVEYING AND NAVIGATION. 


sin. A -f sin. B 
sin. A — sin. B 





Whence, 

sin.^4 -f-sin.i? : sin.^4—sin. B— tan. ^4 . : tan. ^4. -?- 


That is: The sum of the sines of any two arcs is to the dif¬ 
ference of the same sines , as the tangent of one half the sum of 
the same arcs is to the tangent of one half their difference. 

By operating in the same way with the different equations 
in formulae ((7), we find, 


c 


(P)\ 


sin.^l-f sin.i? 

- tan ( A + B 

) (20) 

cos.^4 4- cos.i? 

A 2 i 

sin.^4-(-sin.^ 

- ( A ~ B ' 

) (21) 

cos.i?—cos.^. 

A 2 > 

sin.^4—sin.5 

— tan 1 

) (22) 

cos.^4 + cos.Z? 

A 2 > 

sin. A— sin. B 

— ( A + B ' 

) (23) 

cos.i?—cos.^i 

' 2 > 

cos.JL -fcos.i? 


1 

(24) 

cos.i?—cos.^1' 



\ 2 ) 

1 


These equations are all true, whatever be the value of the 
arcs designated by A and B • we may, therefore, assign any 
possible value to either of them, and if in equations (20), (21), 
and (24), we make _Z> —0, we shall have, 


sin.^L 

— tan ^ 

1 

(25) 

1 + cos.^l 

2 

cot.i A 

sin.J. 

— cot ^ 

1 

(26) 

1— eos..A 

2 

tan. \A 

1 -h cos. A 

_cot.i A 

1 

(27) 

1—cos.J. 

tan. \A 

tan 2 \A 



































TRIGONOMETRY. 


85 


If we now turn back to formulae (ZL), and divide equation 
(X) b y (9)> and (8) by (10), observing at the same time that 

sin 

—1 =tan., w T e shall have, 
cos. 


tan.(#-f&) = 
tan. (#— b) = 


sin .# cosZ + cos.# sin.5 
cos.# cos.5—sin.# sin .b 
sin.# cos. 5—cos.# sin.5 
cos.# cos.5-f-sin.# sin.4 


By dividing the numerators and denominators of the second 
members of these equations by (cos.# cos.J), we find, 


sin.# cos .b . cos.# sin.J 


tan.(#-f &) = 


ta n.(a—b) = 


7 + 

COS.# cos.o 

COS.# COS.5 

COS.# COS.5 

sin.# sinZ 

COS.# COS.5 

cos.# cos.5 

sin.# cosZ 

cos.# sin.5 

cos.# cos.5 

COS.# COS.& 

’ cos.# COS.& 
- + 

sin.# sinZ 


tan.#+tan.5 
1—tan.# tan.5 


(28) 


tan. # —tan. b. 
1+tan.# tan .b 


(29) 


cos.# 


If in equation (11), formulae (Z?), we make a=b, we shall 
have, 

sin.2# = 2 sin.# cos.# (30) 

Making the same hypothesis in equation (13), gives, 

cos.2# + l = 2 cos. 2 # (31) 

The same hypothesis reduces equation (14) to 

1—cos.2# = 2 sin. 2 # (32) 

The same hypothesis reduces equation (28) to 

r\ _ 2 tan.# /oo\ 

tan.2# = -— (33) 

1—tan. 2 # . v 1 

If we substitute # for 2# in (31) and (32), we shall have 

1 + cos.# = 2 cos. 2 !#. (34) 


and 1—cos.# = 2 sin. 2 ^#. 


(35) 





















86 


SURVEYING AND NAVIGATION. 




PROPOSITION III. 


In any right-angled plane triangle , we may have the fol¬ 
lowing proportions : 

1st. The hypotenuse is to either side, as the radius is to the 
sine of the angle opposite to that side. 

2d. One side is to the other side, as the radius is to the tarl' 
gent of the angle adjacent to the first side. 

3d. One side is to the hypotenuse, as the radius is to the 
secant of the angle adjacent to that side. 


Let CAB represent any right- 
angled triangle, right - angled 
at A. 



(Here, and in all cases hereafter, we shall represent the angles of a tri¬ 
angle by the large letters A, B, C , and the sides opposite to them, by the 
small letters a , b, c.) 


From either acute angle, as C, take any distance, as CD, 
greater or less than CB, and describe the arc DE. This arc 
measures the angle C. From D, draw DE parallel to BA • 
and from E, draw EG, also parallel to BA or DF. 

By the definitions of sines, tangents, secants, etc., DF is the 
sine of the angle <7/ EG is the tangent, CG the secant, and 
CF the cosine. 

JSTow, by proportional triangles We have, 

CB : BA — CD : DF; or, a : c = B : sin. (7 

CA : AB — CE: EG ; or, b : c — B : tan. C 

CA : CB — CE : CG ; or, h : a = B : sec. C 

Hence the proposition. 

Scholium.—I f the hypotenuse of a triangle is made radius, one side is the 
sine of the angle opposite to it, and the other side is the cosine of the same 


angle. 


This is obvious from the triangle CDF. 











TRIGONOMETRY. 


87 


PROPOSITION IV. 

In any triangle , the sines of the angles are to one another 
as the sides opposite to them. 

Let ABC be any tri¬ 
angle. From the points 
A and B , as centers, with 
any radius, describe the 
arcs measuring these an¬ 
gles, and draw pa, CD , A a B ' D n B 

and mn, perpendicular to AB. 

Then, 

pec = sin.M, and mn = sin.i?. 

By the similar A’s, Apa and A CD , we have, 

B : sin.ML = h : CD; or B(CD) = b sin.M. (1) 
By the similar a’s, Bmn and BCD , we have, 

B : sin.M? = a : CD; or B(CD) — a si n.B. (2) 

By equating the second members of equations (1) and (2) 

b sin.ML = a sin .B. 

Hence, 

sin.M. : sin.J? = a : 5 

Or, a :b = sin.M. : sin.^. 

Scholium 1.—When either angle is 90°, its sine is radius. 

Scholium 2 .—When CB is less than AC, and the angle B, acute, the tri¬ 
angle is represented by A CB. When the angle B becomes B', it is obtuse, 
and the triangle is A CB '; but the proportion is equally true with either tri¬ 
angle ; for the angle CB'JD — CBA, and the sine of CB'D is the same as 
the sine of AB' C. In practice we can determine which of these triangles is 
proposed, by the side AB being greater or less than AC; or, by the angle 
at the vertex C being large, as A CB, or small, as A CB'. 

In the solitary case in which AC, CB, and the angle A, are given, and 
CB less than A C, we can determine both of the A’si CB and A CB '; and 
then we surely have the right one. 


c 






88 


SURVEYING AND NAVIGATION. 


PROPOSITION V. 

If from any angle of a tria7igle , a perpendicular be let fall 
on the opposite side or base , the tangents of the segments of 
the angle are to each other as the segments of the base. 

Let ABC be the triangle. Let fall 
the perpendicular CD , on the side AB. 

• Take any radius, as Cn, and describe 
the arc which measures the angle C. 

From n , draw qnp parallel to AB. Then 
it is obvious that np is the tangent of the 
angle DCB , and nq is the tangent of the angle A CD. 

How, by reason of the parallels A B and qp, we have, 

qn : np = AD : DB 

That is, tan .ACD : tan .DCB = AD : DB. 

PROPOSITION VI. 

If a perpendicular be let fall from any angle of a triangle 
to its opposite side or base , this base is to the sum of the other 
two sides , as the difference of the sides is to the difference of 
the segments of the base. 

(See Figure to Proposition V.) 

Let AB be the base, and from (7, as a center, with the 
shorter side as radius, describe the circle, cutting AB in G , 
and AC in E\ produce AC to E. 

It is obvious that AE is the sum of the sides A C and CB , 
and AF is their difference. 

Also, AD is one segment of the base made by the perpen¬ 
dicular, and BD = DG is the other ; therefore, the difference 
of the segments is A G. 

As A is a point without a circle, by Cor. Tli. 18, B. Ill, 
we have 

AF x AE = AB x AG 

Hence, AB : AE — AF : A G. 







TRIGONOMETRY. 


89 


PROPOSITION VII. 

The sum of any two sides of a triangle is to their difference , 
as the tangent of one half the sum of the angles opposite to 
these sides , is to the tangent of one half their difference. 


1^ Demonstration. 

Let AJBC he any plane triangle. Then, 
Proposition IY, we have, 

BC : AC — sin.H : sin._Z?. 

Hence, 

BC-\-AC:BC —HC^sin.H-J-sin.J? : sin.H 
But, 


c 



tan. ^ ~ j * tan. ^^ = sin.H -f* sin.j5 : sin.H 

— sin.j5, (eq. ( 19 ), Trig.) 

Comparing the two latter proportions, (Th. 6, B. II), 


we have, 


BC + AC: BC - AC = tan. ± B J : tan. 

Hence the proposition. 

For those, who prefer a demonstration wholly geometrical, 
we give the following: 


2d Demonstration . 


Let ABC be a plane triangle; 
about A as a center, with AB , the 
greater side, as radius describe a 
circle, meeting A C, produced in E 
and F\ and BC in D. Join DA , 
EB, and FB ; and draw FG par¬ 
allel to BC, meeting EB in G . 










I 


90 SURVEYING- AND NAVIGATION. 


The angle EAB is the sum of the angles at the base of the 
triangle ABC (B. I. Th. 12); and EFB at the circum¬ 
ference is equal to one half EAB at the center : EFB is 
therefore equal to one half the sum of the base angles of the 


triangle. 

O 


The angle ACB is equal to CAD and ADC together; or 
since ADC equals ABC\ ACB is equal to the sum of CAD 
and ABC. CAD is therefore equal to the difference between 
the base angles ACB and ABC. Now DBF at the circum 
ference is one half FAD at the center ; and therefore DBF\ 
or its equal (since BC and FG are parallel) BFG is equal to 
one half the difference of the base angles of the triangle. 


Since the angle EBF is inscribed in a semi-circle, it is a 
right angle; and if BF be taken as radius, and a circle 
described from F as a center, BE and BG would be tangents 
(B. Ill, Th. 4); BE being the tangent of the angle EFB , and 
BG of BFG ; or since these angles are the half sum and half 
difference of the base angles of the triangle, BE and BG 
would be tangents of the half sum and half difference of those 
angles. 


From the construction of the figure it is plain that EC equals 
AC+AB and CF equal AB—AC. 

In the triangle EFG , since BC is parallel to FG , we have 
(B. II, Th. 17), 


EC: CF= EB: BG , or 

AB + AC: AB-AC = tan.i (ACB + ABC) : tan A (ACB 

-ABC). 

That is: the sum of two sides is to their difference as the 
tangent of half the sum of the opposite angles is to the tangent 
of half their difference. 




PROPOSITION VIII. 

Given the three sides of any plane triangle , to find some 
relation which they must hear to the cosines of the respective 
angles. 


\ 


TRIGONOMETRY. 


91 


Let ABC be the 
triangle, and let the 
perpendicular fall 
either upon, or with¬ 
out the base, as shown 

in the figures. By <} * D * B 

recurring to Geom. Th. 41, B. I., we shall find 

a 2 + b 2 —c 2 




CD 


2 a 


Now, by Proposition 3, we have 

B : cos. C — b : CD. 
b cos. C 


Therefore, 


CD 


B 


a B x D 


(i) 


( 2 ) 


Equating these two values of CD, and reducing, we have 

(m) 


cos 


Q _ B(a 2 +b 2 —c 2 ) 
%ab 


In this expression we observe, that the part c, whose square 
is found in the numerator with the minus sign, is the side 
opposite to the angle; and that the denominator is twice the 
rectangle of the sides adjacent to the angle. From these 
observations we at once draw the following expressions for the 
cosine A , and cosine B : 


cos. A 


B{b 2 +c 2 —a 2 ) 
%bc 



cos. B = 


B(a 2 +c 2 —b 2 ) 
2 ac 



As these expressions are not convenient for logarithmic com¬ 
putation, we modify them as follows: 

If we put 2 a = A , in equation (31), we have 

cos. A + l = 2cos. 2 ^A. 

In the preceding expression (n), if we consider radius unity, 
and add 1 to both members, we shall have 












92 


SURVEYING AND NAVIGATION. 


cos. A + l = 1 + 


b 2 + e 2 —a 2 


Therefore, 


2 cos.HA = 


2be 

2bc-\-b 2 + c 2 — a 2 


2be 

_ ( b-\-c ) 2 — a 2 
2bc 


Considering b-\-c as one quantity, and observing that ( b-\~ 
c) 2 —a 2 is the difference of two squares , we have 
(b + c) 2 — a 2 = (b + c+a)(b + c—a) ; but (b + e— a)=b+e+a—2a. 

Hence, 2cos. ‘U = ^ + c+a)(i + o+«-2a) 

’ ' 25c 


b-\-c + a\ (b + c+a 


Or, cos. 2 \ A = 


m( 


2 


i 


be 


By putting a ^^ c — s , and extracting square root, the 

2 


final result for radius unity is 

cos 

For any other radius we must write 

cos 


, a _ 4 Ao-«) 

* y 5c 


!.il= 

be 


By inference, cos. \ B — 6 ( f— Q. 

ae 


Also, 


cos 


i (J — a / ^ >3,s '( , s ‘ ~ c ) 

V ab 


In every triangle, the sum of the three angles is equal to 
180°; and if one of the angles is small, the other two must be 
comparatively large; if two of them are small, the third one 
must be large. The greater angle is always opposite the 
greater side; hence, by merely inspecting the given sides, any 
person can decide at once which is the greater angle; and of 
the three preceding equations, that one should be taken which 




















TRIGONOMETRY. 


93 


applies to tlie greater angle, whether that be the particular 
angle required or not; because the equations bring out the 
cosines to the angles; and the cosines to very small arcs vary 
so slowly, that it may be impossible to decide, with sufficient 
numerical accuracy, to what particular arc the cosine belongs. 
For instance, the cosine 9.999999, carried to the table, applies 
to several arcs; and, of course, we should not know which 
one to take; but this difficulty does not exist when the angle 
is large; therefore, compute the largest angle first, and then 
compute the other angles by Proposition IY. 

Equations showing the relations between the sides of a tri¬ 
angle and the sines of the angles, may be readily obtained; 
but as those above given for the cosines in terms of the sides 
are more easily applied and most generally used, we deem them 
sufficient for our purpose. 


OF THE TABLES OF SINES, COSINES, ETC. 

NATURAL SINES, ETC. 

When the radius of the circle is taken as the unit of 
measure, the numerical values of the trigonometrical lines 
belonging to the different arcs of the quadrant, become 
natural sines, cosines, etc. They are then, in fact, but num¬ 
bers expressing the number of times that these lines contain 
the radius of the circle in which they are taken. The tables 
usually contain only the sines and cosines, because these are 
generally sufficient for practical purposes, and the others, 
when required, are readily expressed in terms of them. 

For the method of calculating these functions, the student is 
referred to Kobinson’s New Geometry and Trigonometry, 
page 265 and on. 

Natural sines and cosines are rarely used, except in cases 
where addition or subtraction is to be performed; in all 
case^ of multiplication or division, logarithmic numbers are 


94 


SURVEYING AND NAVIGATION. 


' t , . ' 

preferable, and great pains is taken to adapt the formulae of 
trigonometry to the use of logarithms. 

Natural sines and cosines are given in the tables connected 
with this work, and may be found in connection with loga¬ 
rithmic sines and cosines, in two columns at the right, headed 
“ N. sin.” and “ N. cos.” The degrees under 45° are found at 
the top of the page, those above 45° at the bottom, and the 
minutes either on the left or right hand, as the degrees are 
found at the top or bottom of the page. Natural sines and 
cosines are all calculated with unity as the radius. 

c 

OF LOGARITHMIC SINES, ETC. 

The logarithmic sines and cosines, tangents and cotangents, 
are the logarithms of the number representing these lines in a 
circle whose radius is 10,000,000,000. The radius has a loga¬ 
rithm of 10; and since the trigonometrical lines are propor¬ 
tional to the radii of the circles in which they are calculated, 
the logarithmic sines, &c., may be found by adding 10 to the 
logarithms of the natural sines, &c., as given in the tables. 
The reason why so great a value is assumed for the radius is, 
that when the radius is unity , the sines and cosines are deci- 
malsy and consequently their logarithms have negative indices. 
To avoid this a radius is assumed, whose logarithm 10, added 
to the logarithm of the decimal, will in all ordinary cases 
make the index positive. 

In dealing with logarithmic numbers in trigonometry, 
wherever radius is introduced, its logarithm, 10, must be used ; 
in natural numbers the radius, being unity, is neglected, where 
it is connected with other numbers as a factor. 

The abbreviations sin., cos., tan., cotan., &c., refer to the 
natural numbers; for the logarithmic terms, log. sin., log. 
tan., &c., are used. 

The secants and cosecants of arcs are not given in the table, 
because they are very little used in practice ; and if any par¬ 
ticular secant is required, it can be determined by subtracting 


TRIGONOMETRY. 


95 


the cosine from 20 ; and the cosecant can he found by sub¬ 
tracting the sine from 20. For, 

B 2 B 2 

sec. = -, and cosec. = 

cos. sin. 

The sine of every degree and minute of the quadrant is 
given, directly, in the table, commencing at 0°, and extending 
h) 45°, at the head of the table ; and from 45° to 90°, at the 
bottom of the table, increasing backward. 

The same column that is marked sine at the top, is marked 
cosine at the bottom; and the reason for this is apparent to 
any one who has examined the definitions of sines. 

The difference of two consecutive logarithms is given, cor¬ 
responding to ten seconds. Removing the decimal point one 
figure, will give the difference for one second; and if we mul¬ 
tiply this difference by any proposed number of seconds, we 
shall have a difference of logarithm corresponding to that 
number of seconds above the preceding degree and min¬ 
ute. 

For example, find the sine of 19° IF 22. 

The sine of 19° 17', taken directly from the table, is 9.518829 
The difference for 10" is 60.2; for l", is 6.02; and 

6.02 x 22 = 132 

Hence, the sine of 19° 17' 22" 9.518961 

From this it will be perceived that there is no difficulty in 
obtaining the sine or tangent, cosine or cotangent, of any 
angle greater than 30'. 

Conversely: Given, the logarithmic sine 9.982412, to find 
is corresponding arc. The sine next less in the table is 
9.982404, which gives the arc 73° 48b The difference be¬ 
tween this and the given sine is 8, and the difference for 1" 
is .61; therefore, the number of seconds corresponding to 8, 
must be discovered by dividing 8 by the decimal .61, which 
gives 13. Hence, the arc sought is 73° 48' 13". 

These operations, so similar to those required in the loga¬ 
rithms of simple numbers, will need no rule. 




16 


SURVEYING AND NAVIGATION. 


PRACTICAL APPLICATIONS. 

Having mastered tlie necessary principles, and the explana¬ 
tion of the tables containing the numerical values needed, the 
student may now give his attention to the application of these 
principles and the use of these numbers, in the solution of 
plane triangles. 

I. OF RIGHT-ANGLED TRIANGLES. 

For all the examples which follow, but 
one figure is necessary; and in each case 
the pupil will refer to the one here given, 

AC being the hypotenuse, D the right 
angle, and each angle being represented by the letter at the 
vertex. 

Note. —In all numerical solutions, unless otherwise noted, the reference 
numbers connected with equations and proportions, refer to corresponding 
numbers in the preceding theoretical explanations. 

In every right-angled plane triangle, the right angle being 
always known, there remain five parts: the hypotenuse, base, 
perpendicular, base angle, and perpendicular angle. 

Any two of these being given, the others may be found, 
provided one of the known cpiantities be a side. 



CASE I. 


Given the hypotenuse and an angle . 

(1.) To find the other angle. 

Since A + C ■= 90° .*. 90 °—A — C and 90°— C = A. 


(2.) To find other sides. 

(Prop. Ill) inverting first proportion, 


B : AC = 


j sin. A : CD ) 
t cos.A : AD f 
j sin. C : AD ) 
1 cos. C : CD f 


(1) 

( 2 ) 




TRIGONOMETRY. 


97 


From either the first or second couplet of proportion, CD 
and AD become known. 

EXAMPLES. 

1. Given hypotenuse r= 68, and base angle = 37° 30'. 

90 °—A = 90°—37° 30' = 52° 30' = C ', perpendicular angle. 

By first couplet of proportion above, 

D : 68 : : sin. 37° 30* : CD 
D : 68 : : cos. 37° 30': AD 

Log. 68 - 1.832509 

Log. sin. 37° 30' - 9.784447 

Sum less 10* = 1.616956 

1.616956 = log. CD = log. 41.396. 

Log. 68 = 1.832509 

Log. cos. 37° 30' = 9.899467 

Sum less 10 = 1.731976 

1.731976 = log. AD = log. 53.948. 

(C = 52° 30' 

Ans. ] CD = 41.396 
( AD = 53.948 

2. Given AC — 236, C = 49° 50', to find A, AD and CD. 

(A — 40° 10' 

Ans. 3 AD = 180.34 
( CD = 152.22 

3. Given AC = 92.76 : A — 24° 16', to find other parts. 

tC =65° 44' 

Ans. -j AD — 84.56 
( CD = 38.12 

4. Given AC = 102.8 : A = 42° 48', to find other parts. . 

* The 10 here subtracted is the logarithm of the first term of the pro¬ 
portion. 





/ 


•j$ SURVEYING AND NAVIGATION 


CASE II. 

Given the hypotenuse and one side. 

(1.) To find angles. 

( sin.^1 : CD j 

^ TXT , , a n I cos.-T : AD j ^ 

By Prop. III., as before, B : AC = ( gin c . AJ) 

\ cos. C : CD 

From (1), A may be found; and as before 90°— A—C. Or, 
from (2) <7 may be found ; and 90°— C — A. 


( 2 ) 


(2.) To find other side. 

By Geom., B. I., Th. 39, 

VAC 2 —AID = CD or VAC 2 -CD 2 = AD. (3) 

By Prop. III., 2d proportion, 

B : tan.^i = AD : CD (4) 

B : tan. (7 = CD : AD (5) 

From either Oi which the side reouired may be found. 


EXAMPLES. 

1. Given AC = 100, AD = 48. 

Solution. 

B : 100 = cos.^4 : 48 ' (1) 

Log. B -f log. 48 = 11.681241 

Subtract log. 100 = 2.000000 

-- - - — — 

Log. cos. A — 9.681241 = log. cos. 61° 18' 53" 

A = 61° 18' 53" ; 90°-61° 18' 53" = 28° 41' 7" = C. 

R : tan. 28° 41' 7" = CD : 48 (5) 

Log. R + log. 48 = 11.681241 

Log. tan. 28° 41' 7" = 9.738106 


Log. CD = 1.943134 = log. 87.73. 

Ans. A = 61° 18' 53"; C = 28° 41' 7" ; CD = 87.73. 




X 






TRIGONOMETRY. 


99 



2. Given AC = 300, and CD = 130 

r = 25° 40' 45" 
] <7 = 64° 19' 15" 
( AD = 270.37 

3. Given AC = 1896, and UZ) = 479. 

4. Given AC ■= 241, and ^4Z^ = 78. 


CASE III. 

Given one side and an angle . 

(1.) To find other angle. 

Subtract given angle from 90°. 

(2.) To find hypotenuse. 


By Prop. III., 1st proportion, 

AC : AD — R : sin. C or cos. A (1) 

AC : CD — R : sin. A or cos. C. (2) 

By Prop. III., 3d proportion, 

AD : AC = R : secant A (3) 

CD \ AC — R \ secant C. (4) 


Proportions (1) and (2) would be used most generally, ai 
sines are more easily obtained from table than secants. 

3. To find other side. 

(a) Use AC and angle, and apply Case I. 

(b) Use AC and given side according to Case II. 

(c) By Prop. III., 2d proportion, 

AD : CD = R : tan. A , (5) 

Or, CD : AD — R : tan. C. (6) 


E X A M P L E S. 

1. Given AD = 39, and A = 41° 10'. 
(1.) 90°-41° 10' = 48° 50' = C. 


100 


SURVEYING AND NAVIGATION. 


(2.) A C : 39 = Ti : sin. 48° 50'. (1) 

Log. i?+log. 39 . = 11.591065 

(Less) log. sin. 48° 50' = 9.876678 

Log. AC = 1.714387 = log. 51.81. 

AC = 51.81 

(3.) CD : 39 = It : tan. 48° 50. (6) 

Log. R 4-log. 39 = 11.591065 
Log. tan. 48° 50' = 10.058286 

Log. CD - 1.532779 = log. 34.1 

CD = 34.1. 

i C = 48° 50' 
Am.\ AC = 51.81 
( CD = 34.1 

2. Given CD = 76.84, and A = 51° 42' 20". 

i C =38° 17' 40" 
Am. \ AC = 97.91 
( AD = 60.67. 

3. Given CD = 103.54, and C = 21° 50'. 

I A =68° 10' 
Am. •] AC = 111.54 
( .4i> = 41.48. 

4. Given = 7.96, and C = 17° 23' 12". 

CASE IV. 

r k 

Given the t wo sides or the base and perpendicular. 

(1) To find angles, 

: CD — D : tan. A , or cot. & (1) 

90°-A = (7, or 90°— C = A 

2. To find hypotenuse, 

(a), Apply (2), Case I. 

(Z>), • Apply (2), Case III. 

0), vatd + ssia 


V 





101 


TRIGON OMETRY. 

EXAMPLES. 

1. Given AD = 42.5, and CD = 59. 

/Solution. 

. (1) 12.5 : 59 = R : tan. A. (1) 

Log. 59 +log. R = 11.770852 
Log. 42.5 = 1,62838 9 

Log. tan. A = 10.142463 
10.142463 = log. tan. 54° 14'; 54° 14' = A 

90°—54° 14' = 35° 46' = C. 

(2.) By (2), Case I. 

R : AC = sin. 54° 14' : 59 
Log. 59 +log. A? = 11.770852 
Log. sin. A = 9.909237 

Log. AC = 1.861615 

1.861615 — log. 72.71 .*. AC — 72.71. 

(A = 54° 14' 
Ans. \ C = 35° 46' 
(aC= 72.71 

2. Given AD =. 34.75, and CD = 52.25. 

(i = 56° 22' 24" 
Ans. \c =33° 37' 36" 
(AC =6 2.75 

3. Given AD = 102, and CD = 143. 

r A = 54° 30' 1" 
Ans. \C = 35° 29* 59" 
( AC — 175.65 

4. Given AZ> = 17.377, and CD = 26.89. 

As the solution of right-angled plane triangles is so 
frequently required in practical mathematics, a list of practical 
examples is subjoined, to which the pupil may apply for him¬ 
self the principles and proportions heretofore given. And the 
student will be fully recompensed for all time and labor ex¬ 
pended, by the increased familiarity with these operations. 




102 


SURVEYING AND NAVIGATION. 


PRACTICAL PROBLEMS. 

Let ABC represent any right-angled plane triangle, right- 
angled at B. 

1. In a right-angled triangle, ABC, given the base AB, 
1214, and the angle A , 51° 40' SO'', to find the other parts. 

2. Given A C, 73.26, and the angle A, 49° 12' 20"; required 
the other parts. 

Ans. The angle C, 40° 47' 40"; BC', 55.46 ; and AB, 47.86. 

3. Given AB , 469.34, and the angle A, 51° 26' 17", to find 
the other parts. 

Ans. The angle C, 38° 33' 43"; BC, 588.7; and AC, 752.9. 

4. Given AB, 493, and the angle C, 20° 14'; required, the 
remaining parts. 

Ans. The angle A, 69° 46'; BC, 1337.5 ; and AC, 1425.5. 

5. Let AB = 331, and the angle A =49° 14'; what are 
the other parts ? 

Ans. AC, 506.9; BC, 383.9; and the angle C, 40° 46'. 

6. If AC = 45, and the angle C = 37° 22', what are the 
remaining parts ? 

Ans. AB, 27.31; BC, 35.76; and the angle A, 52° 38'. 

7. Given AC = 4264.3, and the angle A — 56° 29' 13", to 
find the remaining parts. 

Ans. AB, 2354.4; BC, 3555.4; and the angle C, 33°30'47". 

8. If AB = 42.2, and the angle A = 31° 12' 49", what are 
the other parts ? 

Ans. AC, 49.34 ; BC, 25.57; and the angle C, 58° 47' 11". 

9. If AB = 8372.1, and BC — 694.73, what are the other 
parts ? 

Ans. | 8400.9 ; the angle C, 85° 15' 23"; and the 

« angle A, 4° 44' 37". 

10. If AB be 63.4, and AC he 85.72, what are the other 
parts ? 

Ans I 57.69 ; the angle C, 47° 41' 56"; and the angle 
* 1 A, 42° 18' 4". 


N 


TRIGONOMETRY. 103 

11. Given AC = 7269, and AB = 3162, to find the other 
parts. 

\ BC, 6545.23 ; the angle C , 25° 47' 7" ; and the angle 
'( A, 64° 12'53. 

12. Given AC = 4824, and BC — 2412, to find the other 
parts. 

j The angle A = 30° 00', the angle C — 60° OO 
t and Ai? = 4178. 

13. The distance between the earth and sun is 94,770,000 

miles, and at that distance the semi-diameter of the sun sub¬ 
tends an angle of 16' 6". What is the diameter of the sun in 
miles? Ans . 887,673.5 miles. 


B 



In this example, let E be the center of the earth, B that of 
the sun, and EB a tangent to the sun’s surface. Then the 
A EBB is right-angled at B , and BB is the semi-diameter of 
the sun. The value of 2 BB is required. 


II. OF OBLIQUE-ANGLED PLANE TRIANGLES. 

In every obliqne-angled plane triangle, there are six parts; 
viz., three sides and three angles. Any three of these being 
known—provided one side at least be given—the others may 
be found by the preceding theorems and formulae, as is shown 
in the following cases. 

CASE I. 


Given a side and two adjacent angles. 
Let ABC be any plane triangle. 

Suppose AB to be given, and the angles 
A and B. 



A 


B 






104 


SURVEYING AND NAVIGATION. 

(1.) To find third angle. 

By (B. L, Th. 11) 180 °-(AL+jS) = G. 

(2.) To find the other sides. 

By Prop. IY, Sin. C:AB=\ s ! n '„ : B . G > ^ 

" 1 ’ ( sm.B : AO, (b) 

EXAMPLES. 

1. Given AB = 376, A = 48° 3', and B = 40° 14'. 

Solution. 

(1.) 180°—(48° 3'+40° 14’) = 91° 43' = C. 

(2.) Sin. 91° 43' : 376 = sin. 48° 3' : BG. ( a) 

Co. log. sin. 91° 43’ = 0.000195 

Log. 376 = 2.575188 

Log. sin. 48° 3' = 9.871414 

Sum less 10 = 2.446797 

2.446797 = log. 279.77 = BG. 

Sin. 91° 43' : 376 = sin. 40° 14’ : A O (b) 

Co. log. sin. 91° 43' = 0.000195 
Log. 376 = 2.575188 

Log. sin. 40° 14' = 9.810167 

Sum less 10 = 2.385550 

2.385550 - log. 242.97 = AC. 

<C =91° 43'. 
Ans. AC — 242.97. 

( BC = 279.76. 

2. A = 35° 42', B = 76° 27', and AB = 142. 

tC =67° 51 
Ans. \AC= 149.05 
( BC = 89.47 

3. Given B = 23° 40' 22", C = 69° 39’ 51", and BC = 100. 

Ans. 
















TRIGONOMETRY. 


105 


CASE II. 

Given two sides and an angle opposite one of them . 

A 

Let ADC be a plane triangle. Sup¬ 
pose AD and AC to be given, and 
angle D opposite A C. 

(1.) To find other angles. 

By Prop. IY, 

AC : sin. D = AD : sin. angle opposite AD. (1.) 

Now in this proportion the fourth term is a sine; and as the 
sine of an angle and of its supplement are the same, the re¬ 
sult is ambiguous. 

The same ambiguity may be found in the figure; for there 
will be two lines AC, and AE\ one on each side of the per¬ 
pendicular AB , each of which will correspond with the given 
side numericallv. 

Thus, if the angle.Z? be given opposite the shorter side AC, 
there will be two triangles which will contain all the required 
conditions. In practical cases circumstances will generally 
determine which of the two is to be used. 

If the angle given be obtuse, as (7, the other angles must be 
acute, and there can be no ambiguity. 

AC must not be less than AB , the sine of the angle D 
when AD is made radius ; for if so, the triangle becomes im¬ 
possible. 

By the proportion given above having found C and E , we 
have 

180°—(Z>+ C) = DAC, and 180 °-(D+E) = DAE 

(2.) To find third side. 

By Prop. IY, Sin. D : AC = sin. DAC : DC (2) 

Sin. D : AC = sin. DAE : DE (3) 

"When there is no ambiguity, only one of the above will be 
used,—-as only one vertical angle will be found. 






/ 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

1. Given AE = 450, .4(7=309, and E- 27° 50'. 

(1) 309 : sin. 27° 50'=450 : sin. C or E (Eq. 1). 

Co. log. 309 = 7.510041 

Log. sin. 27° 50' = 9.669225 
Log. 450 = 2.653213 

Sum less 10 = 9.832479 = log. sin. C or E 

9.832479 = log. sin. 42° 50' 24" or 137° 9' 36". 

C = 137° 9' 36'', E = 42° 50' 24". 

180°—(27° 50' + 137° 9' 36" = 15° 0' 24" = DA C. (a) 
180°—(27° 50'+ 41° 50'24" = 109° 19' 36" = DAE. (b) 

Sin. 27° 50' : 309 = sin. 15° O' 23" : EC (2) 

Co. log. sin. 27° 50' = 0.330775 
Log. 309 = 2.489959 

Log. sin. 15° 0' 24" = 9.413184 

Sum less 10 = 2.233918 = log. EC. 

2.233918 = log. 171.36 .-. EC = 171.36. 

Sin. 27° 50' : 309 = sin. 109° 19' 36" : EE. 

Co. log. sin. 27° 50' = 0.330775 
Log. 309 = 2.489959 

Log. sin. 109° 19' 36" = 9.974809 

Sum less 10 = 2.795543 = log. EE. 

. 2.795543 = log. 624.52. .-. EE = 624.52. 

( C — 137° 9' 36" ( E =42° 50' 24" 

Ans. (1) \ EAC = 15° 0' 24" (2) \ EAE = 109° 19' 36" 

( EC = 171.36 ( EE = 624.52. 

2. Given, AE = 201, AC = 140, and E = 36° 44'. 

( C = 120° 49'49" ( E = 59° 10' 11" 

Am- ■] EAC = 22°26' 11" \ EAE = 84° 5' 49" 

( EC = 89.34. ( EE = 232.84. 





TRIGON OMETRY. 


107 


3. Given, AD = 180, AC = 100, and C = 127° 33'. 

( D =26° 7' 59" 
Ana. lDAC= 26° 19' 1" 

( DC = 100.65. 

CASE III. 

Given two sides and the angle included by them. 

In the triangle ABC\ let A C and BC be given, 
and the angle C, which they include. 

(1) To find other angles. 

(180°— C) = A+B. 

By Prop. VII, a b 

AC+BC : AC — BC — tan. a (A+B) : tan. | (A—B) (1). 

(A — B) thus becomes known, 

\(A + B) +\(A—B) = A ,) /o\ 

±(A + B)-±(A-B) = B.f 

(2.) To find third side. 

By Prop. IV, 

Sin. A : BC — sin. C : AB , or 

Sin. B : AC = sin. C : AB. ^ 

Which is the greater of the two angles, A and B , is deter¬ 
mined from the sides opposite, which are known ; in the above. 
A is assumed as the greater. 

% 

EXAMPLES. 

1. AC- 97, BC = 113, and G = 63° 41'. 

■ 0 ** ’ 

Solution. 

(1) 180°—63° 41' = 116° 19' = A+B. 

.-. i {A + B) = 58° 9' 30". 

113 + 97 = 210 = AO+BC ; 113-97 = 16 = AC-BC. • 


c 



\ 


/ , 



SURVEYING AND NAVIGATION. 


108 



210 : 16 = tan. 58° 9' 30” : tan. \ (A—B) (1). 

Co. log. 210 = 7.677781 

Log. 16 = 1.204120 

Log. tan. 58° 9' 30” = 10.20688 5 

Sum less 10 = 9.088786 = log. tan. £ (A—B) 

9.088786 = log. tan. 6° 59' 39 '. 

i(A+B) 58° 9' 30” 58° 9' 30” 

\{A — B) + 6° 59' 39” - 6° 59' 39” ^ - 

65° 9' 9” = A 51° 9' 51” = B. 


(2) Sin. 65° 9' 9” : 113 = sin. 63° 41' : AB (3) 

Co. log. sin. 65° 9' 9” = 0.042187 
Log. 113 = 2.053078 

Log. sin. 63° 41' = 9.952481 

Sum less 10 = 2.047746 = log. AB 

2.047746 = log. 111.62 = AB. 

(A =65° 9' 9” 

Ans. \B =51° 9' 51 w 

( AB = 111.62. 

2. Given AB = 100, BC = 69, and B = 31° 30'. 

Ans. A = 41° 12' 36” ; C = 107° 17' 24”; AC = 54.72. 

3. Given AC — 233, BC = 396, and C = 49° 40'. 

4. Given AB = 9.75, AC = 11.5, and At = 70° 11' 10.” 


CASE IV. 

Given the three sides to determine the angles. 

This may be solved by two methods entirely distinct, and 
each solution quite easily obtained. 

ls£ method .—By Prop. VIII, we have formulae for the cosines 
of one-half the angles in terms of the sides. In this case, any 


i 


r 






TRIGONOMETRY. 


109 


of these equations may be used for determining the first angle. 
To avoid danger of ambiguity, arising from the cosines of 
small arcs, it is customary to solve for the largest angle of the 
triangle. The following are the formulae. 


Cos. i A = 

(1) 

iB = 

(2) 

o. J a = 

(3) 


One angle having been obtained, the others may be found 
by Case II. As the largest angle is generally obtained from 
the formula, there is no ambiguity in the other results. 

In these formulae, the letters, a, b, c, represent the sides 
opposite the angles designated by the corresponding capital 
letters. 

2 d method. —In the triangle ABC, draw the perpendicular 
CD, dividing ABC into two right angled 
triangles, 

By Prop. YI, 

AB : BC+ AC=BC-AC : BD-AD . ( 1 ) 

Whence, BD—AD becomes known. b 

l (.AB ) + 2 (. BD—AD ) = BD ; or the 
greater segment, adjacent to the greater side. Also, 

AB-BD = AD. 

In triangle BDC', Prop. Ill, 

BC : BD = B : cos. B. 

In triangle ADC ’ 

A C: AD = B : cos. A. 

180° - (A+B) = a 


( 2 ) 

(3) 

( 4 ) 


c 











110 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

1. Given BC = 122, AC = 107, and AB = 98. 

Solution. — 1st Method. 

Since BC is the longest side, A is the largest angle. Wo 
Use therefore formula (1). 

Cos. Sol = /Bjpi (1) 

£=K a + b + o) - 163.5 ; s —'a = 41.5. 

Log. B 2 = 20. 

Log. 163.5 = 2.213518 
Log. 41.5 = 1.618048 

Log. numerator, 23.831566 

Log. b, 107 = 2.029384 

Log. c, 98 = 1.991226 4.020610 = Log. denom. 

To extract root, 2)19.810956 

Log. cos. I A = 9.905478 = log. 36° 26' 48". 

Therefore, A = 72° 53' 36". 

By Case II, 122 : sin. 72° 53' 36" = 107 : sin. B. 

— 98 : sin. C. 

Co. log. 122 = 7.913640 

Log. sin. 72° 53' 36" = 9.980348 

Log. 107 = 2.029384 . . . j 

Log. sin. B — 9.923372 = log. sin. 56° 57' 20" = B. 

Co. log. 122 — 7.913640 

Log. sin. 72° 53 1 36" = 9.980348 
Log. 98 = 1.991226 

Log. sin. Q — 9.885214 -= log. sin. 50° 9' 4" = G. 

(A = 72° 53' 36" 

} B = 56° 57’ 20" 

( C = 50° 9’ 4". 


Ans. 










TRIGONOMETRY. 




Ill 



/ 


2. Given AB — 214, AG — 176, BC — 200. 

Solution .—2 d Method. 

214 : 376 = 24 : BD - AD. (1) 

Log. 376 = 2.575188 

Log. 24 = 1.380211 

Subtract log. 214 = 2.330414 

Log. (. BD - AD),= 1.624985; .-. BD-AD = 42.168. 
|(214 -f 42.168) = 128.084 = BD, adjacent to BC. 
^(214 — 42.168) = 85.916 = AD. 

In triangle BD C, 

200 : 128.084 = Ii : cos. B. (2) 

Co. log. 200 = 7.698970 

Log. 128.084 = 2.107495 ' 

Sum-flog. R —10 = 9.806465 = log. cos. B. 

Whence, B = 50° 10' 37". 

In triangle ADC, 

176 : 85.916 = R : cos. A. (3) 

Co. log. 176 = 7.754487 

Log. 85.916 = 1.934074 

Sum-flog. R —10 -' 9.688561 = log. cos. A. 

Whence, A = 60° 46' 49". 

180°—110° 57' 26" = 69° 2' 34" = C. 

(A = 60° 46' 49" 
Am. <B = 50° 10' 37" 
( C = 69° 2' 34" 

3. Given AB = 76.5, AC - 51.75, and BC = 43.25. 

. A = 32° 44' 31" 
Arts. \ B = 40° 19' 37" 
( C = 106° 55 1 52" 






112 


SURVEYING AND NAVIGATION. 


4. Giverj AB = 34, AC = 48, and BO = 30. 

r M, 38° 20' 33" 
Am. ] B, 96° 58' 57" 

( C, 44° 40' 30" 

Having given the theory and practice of the solution of 
oblique-angled plane triangles, we add the following exam¬ 
ples, where the student will select and apply the proper 
methods to be employed. 


PRACTICAL PROBLEMS. 

Let ABC represent any oblique-angled triangle. 

1. Given AB 697, the angle A 81° 30' 10", and the angle 
B 40° 30' 44", to find the other parts. 

Am. AC, 534; BC, 813 ; and L C, 57° 59' 6". 

2. If AC - 720.8, l A = 70° 5' 22", \_B = 59° 35'36", 
required the other parts. 

Am. AB , 643.2; BC, 785.8 ; and L C , 50° 19' 2". 

3. Given BC 980.1, the angle A 7° 6' 26", and the angle 
B 106° 2' 23", to find the other parts. 

'Am. AB , 7283.8; AC, 7613.1 ; and L C, 66° 51' 11". 

4. Given AB 896.2, BC 328.4, and the angle C 113° 45 r 
20", to find the other parts. 

Ans. AC ; 712 ; L A, 19° 35' 46" ; and L B , 46° 38' 54". 

5. Given AC — 4627, BC — 5169, and the angle A =*. 
70° 25' 12", to find the other parts. 

Ans. AB , 4328 ; L B , 57° 29' 56"; and L C, 52° 4' 52". 

6. Given AB 793.8, BC 481.6, and AC 500.0, to find the 
angles. 


Ans \ L 35° 15'32"; [_ B, 36° 49' 18"; 
1 and L C, 107° 55' 10'. 


TRIGONOMETRY. 


113 


7. Given AB 100.3, BC 100.3, and AC 100.3, to find the 
angles. 

An8 j The angle A, 60° ; the angle B, 
} 60° ; and the angle C,, 00°. 

8. Given AB 92.6, BC 46.3, and AC 71.2, to find the 
angles. 

Ans ( L ^4,29°17'22"; |_ 5,48°47'30"; 
I and L C, 101° 55' 8". 

9. Given AB 4963, BC 5124, and AC 5621, to find the 
angles. 

42' 36"; 


Ans { A 57° 30' 2S" ; L B , 67< 
( and L C, 54° 46' 55". 


10. Given AB 728.1, BC 614.7, and AC 583.8, to find the 
angles. 

A j L A = 54° 32' 52", [_ B = 50° 40' 58", 
I and L C= 74° 46' 10". 

11. Given AB 96.74, BC 83.29, and AC 111.42, to find 
the angles. 

j L A = 46° 30' 45", L B = 76° 3' 46", 
| and L C = 57° 25' 29". 


12. Given AB 363.4, BC 148.4, and the angle B 102° 18 r 
27", to find the other parts. 

A ( L A = 20° 9' 17", the side AC = 420.8, 
* \ and L O = 57° 32' 16" 

13. Given AB 632, BC 494, and the angle A 20° 16’, to 
find the other parts, the angle C being acute. 

. j L C = 26° 18' 19", L B— 133° 25 f 41", 
AnS - | and AG = 1035.7. 

14. Given AB 53.9, AC 46.21, and the anglei? 58° 16', *o 
find the other parts. 

Ans. L A = 38° 58', L C = 82° 46', and = 34.16. 


114 


SURVEYING AND NAVIGATION. 


15. Given AB 2163, BC 1672, and the angle C 112° 18' 22", 
to find the other parts. 

Ans. AO, 877.2 ; L B, 22° 2' 16" ; and L A, 45° 39' 22". 

16. Given AB 496, BC 496, and the angle B 38° 16' to find 
the other parts. 

Ans. AC, 325.1; \_A, 70° 52'; and L C, 70° 52'. 

% 

17. Given AB 428, the angle C 49° 16', and {AC+BC) 
918, to find the other parts, the angle B being obtuse. 

. j The angle A = 38° 44' 48", the angle B = 91° 

S ' \ 59' 12", AC= 564.49, and BC= 353.5. 

18. Given AC 126, the angle B 29° 46', and (AB—BC) 
43, to find the other parts. 

j \ The angle A = 55° 51' 32", the angle C = 94° 
nS ‘ \ 22' 28", AB = 253.05, and BC= 210.054. 

19. Given AB 1269, AC 1837, and the angle A, 53° 16' 20", 
to find the other parts. 

j L B - 83° 23' 47", L C = 43° 19' 53", and BC 
( =1482.16. 


SECTION III, 

OF SPHERICAL TRIGONOMETRY. 

A Spherical Triangle contains six parts—three sides and 
three angles—any three of which being given, the other three 
may be determined. 

Spherical Trigonometry has for its object to explain the 
different methods of computing three of the six parts of a 
spherical triangle, when the other three are given. It may 
be divided into Right-angled Spherical Trigonometry, and 

V ' - v S 


I 



\ 


TRIGONOMETRY. H5 

Oblique angled Spherical Trigonometry; the first treating of 
the solution of right-angled, and the second of oblique-angled 
spherical triangles. 


RIGHT-ANGLED SPHERICAL TRIGONOMETRY. 



PROPOSITION I. 

With the shies of the sides , and the tangent of one side oj 
any right-angled spherical triangle , two plane triangles can be 
formed that will be similar , and similarly situated . 

Let ABC be a spherical triangle, right- 
angled at Id * and let D be the center of 
the sphere. Because the angle CBA is a 
right angle, the plane CBD is perpendic¬ 
ular to the plane DBA. From C let fall ^ 

CII, perpendicular to the plane DBA; 
and as the plane CBD is perpendicular to 
the plane DBA , CH will lie in the plane CBD , and be per¬ 
pendicular to the line DB , and perpendicular to all lines that 
can be drawn in the plane DBA , from the point H (Def. 2, 
B. VI. Geom.) 

Draw HG perpendicular to DA, and draw GC / GC will 
lie wholly in the plane CD A, and CIIG is a right-angled trb 
angle, right-angled at H. 

We will now demonstrate that the angle DGC is a right 
angle. 

The right-angled aCHG, gives CID -4- HG' = CG' (1) 

The right-angled A DGH, gives DG' + IIG' = DH' (2) 

By subtraction, CID — DG' = CG* — DID (3) 

By transposition, CID + DH' = CG* + DG' (4) 

But the first member of equation (4) is equal to CD', be¬ 
cause CDH is a right-angled triangle; 

Therefore, CD' = CG' + DG' 

Hence, CD is the hypotenuse of the right-angled triangle 
DGC , (Th. 39, B. I. Geom.) 





116 SURVEYING AND NAVIGATION. 

From the point B, draw BE at right angles to DA, and 
BE at right angles to DB , in the plane CDB extended ; the 
point F will be in the line DC. Draw EE\ and as F is in 
the plane CD A, and E is in the same plane, the line EF is in 
the plane CD A. Now we are to prove that the triangle CIIG 
is similar to the triangle BEE\ and similarly situated. 

As IIC and BE are both at right angles to DA y they are 
parallel; and as HC and BE are both at right angles to DB , 
they are parallel; and by reason of the parallels, the angles 
GIIC and EBF are equal; but GIIC is a right angle ; there¬ 
fore, EBF is also a right angle. 

Now, as GH and BE are parallel, and CII and BE are 
also parallel, Ave have, 

DR : DB = IIG : BE 
And, DR:DB = HC: BE 

Therefore, IIG : BE = 11C : BE (Th. 6, B. II.), 

Or, RG : IIC = BE : BE. 

Here, then, are two triangles, having an angle in the one 
equal to an angle in the other, and the sides about the equal 
angles proportional; the t\\ T o triangles are therefore equiangu¬ 
lar (Geom. Cor. 2, Th. 17, B. II.); and they are similarly 
situated, for their sides make equal angles at AT and B witli 
the same line, DB. 

Hence the proposition. 

Scholium — By the definition of sines, cosines, and tangents, we ner- 
ceive that CH is the sine of the arc BC , DH is its cosine, and BF its tan¬ 
gent; CG is the sine of the arc AC, and DG its cosine. Also, BE is the 
sine of the arc AB, and BE is the cosine of the same arc. With this figure 
we are prepared to demonstrate the following propositions. 

PROPOSITION II. 

In any right-angled spherical triangle , the sine of one side 
is to the tangent of the other side , as radius is to the tangent of 
the angle adjacent to the first-mentioned side. 

Or, the sine of one side is to the tangent of the other side , as 


TRIGONOMETRY. 


117 


the cotangent of the angle adjacent to the first-mentioned side 
is to the radius . 

For tlie sake of brevity, we will represent the angles of the 
triangle by A, B , C ; and the sides or arcs opposite to these 
angles by a, 5, c ; that is, a opposite A, etc. 

In the right-angled plane triangle EBF , we have, 

: BF — B : tan .BEF 
That is, sin.c : tan. <2 = B : t&n.A, 

which agrees with the first part of the enunciation. By refer¬ 
ence to equation (5), Plane Trigonometry, we shall find that 


t&n.A cot.^L = B 2 ; 

B 2 

therefore, tailed. = ---. 

cot.^1 

Substituting this value for tangent A, in the preceding pro¬ 
portion, and dividing the last couplet by B, we shall have, 


sin.<? : tan .a = 


1 : 


B 

cot.^I 


: Or, sin.c : tan .a — cot. A : B. 

Or, B sin.c = tan.& cot.-T, (1) 

which answers to the second part of the enunciation. 

Cor. By changing the construction, drawing the tangent to 
AB in place of the tangent to i?(7,and proceeding in a similar 
manner, we have, 

B sin.& = tan.c cot .C. (2) 


PROPOSITION III. 

In any right-angled spherical triangle , the sine of the right 
angle is to the sine of the hypotenuse , as the sine of either of 
the other angles is to the sine of the side opposite to that angle. 

The sine of 90°, or radius, is designated by B. 

In the plane triangle CIIC , we have, 

sin. CHG : CO = sin .COB: CE 

B : sin.J = sin.^1 : sin .0 
B sin.& = sin.J sin.^I. 


That is, 
Or, 


( 3 ) 




118 


SURVEYING- AND NAVIGATION. 

/ 

Cor. By a change in the construction of the figure, drawing 
a tangent to AB , etc., we shall have, 

B : sin .b — sin. C : sin.c 

Or, B si 11.0 = sin .b sin. (7. (4) 


Scholium. —Collecting the four equations taken from this and the prece¬ 
ding proposition, \Ve have, 


(1) R sin.c = tan.a cot.A 

(2) R sin.a = tan.c cot. C 

(3) R sin.a = sin .b sin.A 

(4) R sin.c = sin .b sin. C , 

These equations refer to the right-angled triangle, 
ABC ; but the principles are true for any right- 
angled spherical triangle. Let us apply them to the 
right-angled triangle, PCD, the complemental tri¬ 
angle to ABC. 

Making this application, equation 



P' 


(1) becomes R sin. CD = tan .PD cot. C ( n ) 

(2) becomes R sin .PD = tan. CD cot .P (m) 

(3) becomes R sin .PD = sin. PC sin. C ( ° ) 

(4) becomes R sin .CD = sin.PC? sin.P ( p ) 


By observing that sin. CD = cos. A C = cos A, and that ia.ii.PD — cot .DO 
= cot.A, etc.; and by running equations (n), (m), (o'), and (ph back 
into the triangle ABC\ we shall have, 


(5) R cos .b = cot.A cot. C' 

(6) R cos.A = cot .b tan.c 

(7) R cos.A = cos.a sin. C 

(8) R cos .b = cos.a cos.c 

By observing equation (6), we find that the second member refers to 
sides adjacent to the angle A. The same relation holds in respect to the 
angle C, and gives, 

(9) R cos. C = cot. b. tan.a. 

• Making the same observations on (7), we infer, 

(10) R cos. C = cos.c sin.A. 

Observation 1.—Several of these equations can be deduced 
geometrically without the least difficulty. For example, tako 








TRIGONOMETRY. 


119 


tlie figure to Proposition I. The parallels in the plane DBA 
give 

DB : DII — DE\ DG. 

That is, B : cos .# — cos.# : cos.5. 

A result identical with equation (8), and in words it is ex¬ 
pressed thus : Badius is to cosine of one side , as the cosine of 
he other' side is to the cosine of the hypotenuse. 

Observation 2.—The equations numbered from (1) to (10) 
cover every possible case that can occur in right-angled spheri¬ 
cal trigonometry ; but the combinations are too various to be 
remembered, and readily applied to practical use. 

We can remedy this inconvenience, by talcing the comple¬ 
ment of the hypotenuse, and the complements of the two 
oblique angles, in place of the arcs themselves. 

Thus, b is the hypotenuse, and let V be its complement. 

Then, b + V — 90°; or, b = 90°—^; and, sin.J = cos.£ f , 
cos .b = sin.Z»'; tan. b = cot.£ f . 

In the same manner, if A 1 is the complement to A, then 
sin.A = cos.A'; cos.A = sin.A' ; and, tan.A = cot.A' ; and 
similarly, sin.6 7 = cos. C ; cos.6 7 = sin. 6"; and tan.f7 = cot .C l . 

Substituting these values for b , A , and { 7 , in the foregoing 
ten equations (# and c remaining the same), we have, 


(11) 

( 12 ) 

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 

(19) 

( 20 ) 


NAPIER’S CIRCIJ 

B sin.# = tan.# tan.A' 

A sin.# — tan.# tan. C l 
B sin.# = cos.^ cos.A' 

B sin.# = cos.Z» f cos.6 17 
B smB = tan.A' tan.6 7 
B sin.A* — tan .V tan.# 

B sin.A' = cos.# cos .C l 
B smB = cos.# cos.# 

B sin.6 7f = tan.Z»' tan.# 

B sin .C 1 = cos.#' cos.A' 


A R PARTS. 

Omitting the consid¬ 
eration of the right an¬ 
gle, there are five parts. 
Each part taken as a 
middle part, is connect¬ 
ed to its adjacent parts 
by one equation, and to 
its extreme parts by an¬ 
other equation; there¬ 
fore ten equations are 
required for the com¬ 
binations of all the parts. 



120 


SURVEYING AND NAVIGATION. 


These equations are very remarkable, because the first mem¬ 
bers are all composed of radius into some sine , and the second 
members are all composed of the product of two tangents , or 
two cosines. 

To condense these equations into words, for the purpose of 
assisting the memory, we will refer any one of them directly 
to the right-angled triangle, ABC\ in the last figure. 

When the right-angle is left out of the question, a right- 
angled triangle consists of five parts— three sides, and two 
angles. Let any one of these parts be called a middle part j 
then two other parts will lie adjacent to this part, and two 
opposite to it , that is, separated from it by two other parts. 

For instance, take equation (11), and call ctlie middle part j 
then A' and a will be adjacent parts, and C 1 and V opposite 
parts. Again, take <zasa middle part / then c and C will be 
adjacent parts, and A' and V will be opposite parts; and thus 
we may go round the triangle. 

Take any equation from (11) to (20), and consider the mid¬ 
dle part in the first member of the equation, and we shall find 
that it corresponds to one of the following invariable and 
comprehensive 7'ules. 

1. The radius into the sine of the middle part is equal to 
the product of the tangents of the adjacent parts. 

2. The radius into the sine of the middle part is equal to 
the product of the cosines of the opposite parts. 

These rules are known as Napier’s Rules, because they were 
first given by that distinguished mathematician, who was also 
the inventor of logarithms. 

In the application of these equations, the accent may be 
omitted if tan. be changed to cotan., sin. to cosin., etc. Thus, 
if equation (13) were to be employed, it would be written, in 
the first instance, R sin.& = cos.& f cos.A', to insure conformity 
to the rule; then, we would change it into R sin. a — sin .b 
sin .A. 

\ ” 

Remark. —We caution the pupil to be very particular to take the comple¬ 
ments of the hypotenuse, and the complements of the oblique angles. 


TRIGONOMETRY. 


121 


SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES. 

A good general conception of the sphere is essential to a 
practical knowledge of spherical trigonometry, and this con¬ 
ception is best obtained by the examination of an artificial 
globe. By tracing out upon its surface the various forms of 
right-angled and oblique-angled triangles, and viewing them 
from different points, we may soon acquire the power of mak¬ 
ing a natural representation of them on paper, which will be 
found of much assistance in the solution and interpretation of 
problems. 

For instance, suppose one side of a right-angled spherical 
triangle to be 56°, and the angle between this side and the 
hypotenuse to be 24°. AVliat is the hypotenuse, and what the 
other side and angle % 

A person might solve this problem by the application of the 
proper equations or proportions, without really comprehending 
it; that is, without being able to form a distinct notion of the 
shape of the triangle, and of its relation to the surface of the 
sphere on which it is situated. 

If we refer this triangle to the common geographical globe, 
the side 56° maybe laid off on the equator, or on the meridian. 
In the first case, the hypotenuse will be the arc of a great cir¬ 
cle drawn through one extremity of the side 56°, above or 
below the equator, and making with it an angle of 24°; the 
other side will be an arc of a meridian. In the second case, 
the side 56° falling on a meridian, the hypotenuse will be the 
arc of a great circle drawn through one extremity of this side, 
on the right or left of the meridian, and making with it an 
angle of 24° ; the other side will be the arc of a great circle, 
at right angles to the meridian in which the given side lies. 

Generally speaking, the apparent form of a spherical tri¬ 
angle, and consequently the manner of representing it on 
paper, will differ with the position assumed for the eye in 

viewing it. From whatever point we look at a sphere, its 

6 


122 


SURVEYING AND NAVIGATION. 



uated ; and when the eye is, as will be hereafter supposed, at 


an infinite distance, this circle will he a great circle of the 
sphere. All great circles of the sphere whose planes pass 
through the eye, will seem to be diameters of the circle which 


represents the outline of the sphere. 

We will now suppose the eye to be in the plane of the equa¬ 
tor, and proceed to construct our triangle on paper. 

Let the great circle, PASA', 


p 


represent the outline of the 
sphere, the diameter AA 1 the 
equator, and the diameter PS 
the central meridian, or the 



meridian in whose plane the a 


A 


eye is situated. Let AB=56° t / / 

represent the given side, and \ \ I / 

AO, making with ABthv angle \. \ / / 

BAC = 24°, the hypotenuse; 
then will BO , the arc of a 8 

meridian, be the other side at right angles to AB , and the tri¬ 
angle, ABO, corresponds in all respects to the given triangle. 

Again, measure off 56° from P to Q, draw the arc DQ, 
make the arc A’G equal to 24°, and draw the quadrant PPG. 
The triangle PQP will also represent the given triangle in 
every particular. 

We know from the construction that BY — 24° is greater 
than BO, and that AO is greater than AB, that is, greater 
than 56°. 

In like manner, we know that A' = 24° is greater than QP, 
ancl that PP is greater than PQ, because PP is more nearly 
equal to PG = 90° than PQ is to PA ~ 90. 

For illustration and explanation, we also give the following 
example: 

In a right-angled spherical triangle there are given the 
hypotenuse equal to 150° 33' 20", the angle at the base 23° 
27' 29", to find the base and the perpendicular. Let A'BO 






TRIGONOMETRY. 


123 


in the last figure, represent the triangle in which A 1 C = 150° 
33' 20", the L BA'C = 23° 27' 29", and the sides A'B and 
BC are required. 

This problem presents a right-angled spherical triangle, 
whose base and hypotenuse are each greater than 90°; and in 
cases of this kind, let the pupil observe, that the base is greater 
than the iiyjjotenuse , and the oblique angle opposite the base, 
is greater than a right angle. In all cases, a spherical triangle 
and its supplemental triangle make a lune. It is 180° from 
one pole to its opposite, whatever great circle be traversed. 
It is 180° along the equator, ABA ', and also 180° along the 
* ecliptic AC A 1 . The lune always gives two triangles; and 
when the sides of one of them are greater than 90°, we take 
the triangle having supplemental sides ; hence in this case we 
operate on the triangle ABC. 

AC is greater than AB , therefore A'B is greater than the 
hypotenuse A'C. 

The L ACB is less than 90°; therefore the adjacent angle 
A ] CB is greater than 90°, the two together being equal to 
two right angles. 

When a side and opposite angle are both greater or both 
less than 90°, they are said to be of the same affection • when 
the one is greater and the other less than 90°, they are said to 
be of different affection. 

Now, if the two sides of a right-angled spherical triangle are 
of the same affection , the hypotenuse will be less than 90°; 
and if of different affection , the hypotenuse will be greater 
than 90°. 

If, in every instance, we make a natural construction of the 
figure, and use common judgment, it will be impossible to 
doubt whether an arc must be taken greater or less than 90°. 

We will now solve the triangle ACB. ^16—180°—150° 
33' 20" = 29° 26' 40". 

To find BC , we use Eq. (3) or (13), Prop. III., thus: 


124 SURVEYING AND NAVIGATION. 

b, sin. 29° 20’ 40" 9.691594 

A, sin. 23° 27' 29" 9.599984 

a, sin. 11° 17’ 7" 9.291578 

To find AB, we use equation (1) or (11), thus: 

a, tan. 11° IT' 7" 9.300016 

A, cot. 23° 27' 29" 10.362674 

c, sin. 27° 22' 32" 9.662690 

180 

A'B = 152° 37' 28" 

Remark.— The small letters given in the preceding equations correspond 
to the sides opposite the angles of like letters. 

The student should familiarize himself thoroughly with the rules of Napier 
by careful solutions of the following 


PRACTICAL PROBLEMS IN RIGHT-ANGLED SPHERICAL 

TRIGONOMETRY. 

1. In the right-angled spherical triangle 
ABC, given AB = 118° 21' 4", and the 
angle A = 23° 40' 12", to find the other 
parts. 

£ ns ( AO, 116° 17' 55"; the angle C 100° 59' 26"; and 
' i BO ; 21° 5' 42". 

2. In the right-angled spherical triangle ABC, given AB 
53° 14' 20", and the angle A 91° 25' 53", to find the other 
parts. 

j^ ns j AC, 91° 4' 9"; the angle C, 53° 15' 8"; and BC, 

* ( 91° 47' 10". 

3. In the right-angled spherical triangle ABC, given AB 
102° 50' 25", and the angle A 113° 14' 37", to find the other 
parts. 

A ns j AC, 34° 51' 36"; the angle C, 101° 46' 57"; and 

* 1 BC, 113° 46' 27". 









TRIGONOMETRY. 


125 


4. In the right-angled spherical triangle ABC \ given AB 
4S° 24' 16", and BC 59° 38' 27", to find the other parts. 

A ( AC, i 70° 23' 42"; the angle A, 66° 20' 40" ; and the 
1 angle C, 52° 32' 56". 

5. In the right-angled spherical triangle ABC , given AB 
151° 23' 9", and BC 16° 35' 14", to find the other parts. 

An$ | AC, 147° 16' 51"; the angle C, 117° 37' 21"; and 
1 the angle A, 31° 52' 49". 

6. In the right-angled spherical triangle ABC ' given AB 
73° 4' 31", and AC 86° 12' 15", to find the other parts. 

Ans i 76° 51 ; 20" ; the angle A , 77° 24' 23" ; and the 
. ( angle C , 73° 29' 40". 

7. In the right-angled spherical triangle ABC, given AC 
118° 32' 12", and AB 47° 26' 35", to find the other parts. 

a ( BC, 134° 56' 20"; the angle A, 126° 19' 2"; and the 
1 angle C, 56° 58' 44". 

8. In the right-angled spherical triangle ABC, given AB 
40° 18' 23", and AC 100° 3' 7", to find the other parts. 

Ans j The angle A, 98° 38' 53"; the angle C, 41° 4' 6"; 
’ l and BC, 103° 13' 52". 


9. In the right-angled spherical triangle ABC, given AC 
61° 3' 22", and the angle A 49° 28' 12", to find the other 
parts. 


Ans. 


AB, 49° 36' 6"; the angle C, 60° 29' 20"; and BC, 
41° 41' 32". 


10. In the right-angled spherical triangle ABC, given AB 
29° 12' 50", and the angle C 37° 26' 21", to find the other 
parts. 

r Ambiguous; the angle A, 65° 27' 57", or its supple- 
Ans.\ ment; AC, 53° 24' 13", or its supplement; BC, 
( 46° 55' 2 ", or its supplement. 

11. In the right-angled spherical triangle ABC, given AB 


126 


SURVEYING AND NAVIGATION. 


100° 10' 3", and the angle C 90° 14' 20", to find the other 
parts. 

r AC, 100° 9' 52", or its supplement; BC, 1° 19' 55", 
Ans. ■] or its supplement; and the angle A, 1° 21' 12', or 
( its supplement. 

12. In the right-angled spherical triangle ABC', given AB 
54° 21' 35', and the angle C 61° 2' 15", to find the other parts. 

/ BC, 129° 28' 28", or its supplement; AC, 111° 44' 
Ans. -! 34", or its supplement; and the angle A, 123° 47' 

( 44", or its supplement. 

13. In the right-angled spherical triangle ABC, given AB 
121° 26' 25", and the angle C 111° 14' 37", to find the other 

parts. . 

( The angle A, 136° 0' 5", or its supplement; AC, 66° 
Ans. i 15' 38", or its supplement; and BC, 140° 30' 57", 
( or its supplement. 


QUADRANTAL TRIANGLES. 



The solution of right-angled spherical 
triangles Includes, also, the solution of 
quadrantal triangles, as may be seen by 
inspecting the adjoining figure. When we A | 
have one quadrantal triangle we have four, 
which, with one right-angled triangle, fill 
up the whole hemisphere. 

To effect the solution of either of the four quadrantal tri¬ 
angles, ABC, ABC, A'PC, or ABC, it is sufficient to solve 
the small right-angled spherical triangle ABC. 

To the half lime ABB, we add the triangle ABC, and we 
have the quadrantal triangle AP'C; and by subtracting the 
same from the equal half lune APB, we have the quadrantal 
triangle PAC. 

When we have the side AC, of the same triangle, we have 
its supplement AC, which is a side of the triangles A PC, 






TRIGONOMETRY. 


127 


and A’PC. When we have the side CB , of the small tri- 
ungie, by adding it to 90° we have P C\ a side of the triangle 
A'P 1 Cy and subtracting it from 90°, we have jPC, a side of 
the triangles AFC , and A 1 PC . 


PROBLEM I. 

In a quadrantal triangle , there are given the quadrantal 
side 90°, a side adjacent 42° 21', and the angle opposite this 
last side , equal to 36° 31'. Required the other parts. 

By this enunciation we cannot decide whether the triangle 
APC or APC is the one required, for AC—^P 21' belongs 
equally to both triangles. The angle APC—AP 1 C=2>§° 3B 
= AB. 

We operate wholly on the triangle ABC. To find the angle 
A , call it the middle part. Then, 

R cos.CAB = R sin .PAC — cot .AC t&n.AB. 

cot .AC = 42° 21' 10.040231 

tsm.AB = 36° 31' 9.869473 

cos. CAB = 35° 40' 51" 9.909704 

90° 

PAC = 54° 19' 9" 

PAC - 125° 40' 51" 

To find the angle C\ call it the middle part. Then, 

R cos .ACB — An.CAB cos .AB. 

sin .CAB = 35° 40' 51" 9.765869 

cos .AB = 36° 31' 9.905085 

cos.ACB = 62° 2' 45" 9.570954 

180° 

A CP — A ] CP~^m° 57' 15" 






128 


SURVEYING AND NAVIGATION. 


To find the side £C> call it the middle part. Then, 

sin .£0 = tsn\.A£ cot.AC£. 

tm.A£ = 36° 31' 0" 9.869473 

eotJLCB = 62° 2' 45" 9.724835 

&m.£C = 23° 8' 11" 9.594308 

90° 

PC = 66° 51' 49" 

P } G = 113° 8' 11" 

We now have all the sides, and all the angles of the fcmr 
triangles in question. 


PROBLEM II. 

In a quadrantal spherical triangle , having given the quad- 
rantal side 90°, an adjacent side 115° 9', and the included 
angle 115° 55', to jind the other parts. 

This enunciation clearly points out the 
particular triangle A'PC. AP l — 90°; 
and conceive A'C—lli) 0 9*. Then the 
angle PM'£7=115° 55 '=£'£. 

From the angle PA'C take 90°, or 
PA'£ , and the remainder is the angle 
OA , £=£AC= 25° 55'. 

We here again operate on the triangle A£C. A ! C taken 
from 180°, gives 64° 51' = AC. 

To find £C ‘ we call it the middlepart. Then, 

Bsm.£C = sm.AC &m.£AC. 

m\.AC = 64° 51' 9.956744 

&m.£AC = 25° 55' 9.640544 

&m.£C = 23° 18' 19" 9.597288 

90° 


p 



P}C = 113° 18' 19" 









TRIGONOMETRY. 


129 


To find AB , we call it the middle part. Then, 

B sin .AB = tan.BC cot.BAC. 

tan.B C = 23° 18' 19" 9.634251 

cot.BAC = 25° 55' 9.313423 

sin.AB = 62° 26' 8" 8.947674 

180° 

A'B = 117° 33' 52" = the angle A’B'C. 

To find the angle O, we call it the middle part. Then, 
B cos.6 7 = dot.AC tsm.BC. 

cot.. AC — 64° 51' 9.671634 

tan .BC — 23° 18' 19” 9.634251 

cos. C = 78° 9.305885 

180° 19' 53” 

B’CA’ = 101° 40' 7” 

Thus, we have found the side B'C = 113° 18' 19” \ 

The angle A'P'C = 117° 33' 52” [ Ans. 
“ P'CA' = 101° 40' 7” ) 


PRACTICAL PROBLEMS. 

1. In a quadrantal triangle, given the quadrantal side, 90°, a 
side adjacent, 67° 3', and the included angle, 49° 18', to find 
the other parts. 

t The remaining side is 53° 5' 44”; the angle opposite 
Ans. -j the quadrantal side, 108° 32' 29”; and the remain- 

( ing angle, 60° 48' 54”. 

2. In a quadrantal triangle, given the quadrantal side, 90°, 
one angle adjacent, 118° 40' 36”, and the side opposite this 
last-mentioned angle, 113° 2' 28”, to find the other parts. 

( The remaining side is 54° 38' 57”; the angle oppo- 
Ans. a site, 51° 2' 35”; and the angle opposite the quad- 

( rantal side, 72° 26' 21”. 

9 







130 


i 


SURVEYING AND NAVIGATION. 

3. In a quadrantal triangle, given the quadrantal side, 90°, 
and the two adjacent angles, one 69° 13' 40", the other 72° 12' 
4", to find the other parts. 

/ One of the remaining sides is 70° 8' 39", the other is 

Ans. 73° 17' 29", and the angle opposite the quadrantal 
( side is 96° 13' 23". 

4. In a quadrantal triangle, given the quadrantal side, 90°, 
one adjacent side, 86° 14' 40", and the angle opposite to that 
side, 37° 12' 20", to find the other parts. 

( The remaining side is 4° 43' 2"; the angle opposite, 

Ans. ■< 2° 51' 23" ; and the angle opposite the quadrantal 

( side, 142° 42' 2". 

5. In a quadrantal triangle, given the quadrantal side, 90°, 
and the other two sides, one 118° 32' 16", the other 67° 48' 
40", to find the other parts—the three angles. 

j The angles are 64° 32' 21", 121° 3' 40", and 77° 11' 

Ans. •<’ 6" ; the greater angle opposite the greater side, of 

( course. 

6. In a quadrantal triangle, given the quadrantal side, 90°, 
the angle opposite, 104° 41' 17", and one adjacent side, 73° 21' 
6", to find the other parts. ' 

m j Remaining side, 49° 42' 16"; remaining angles, 47° 
m ' \ 32' 38", and 67° 56' 13". 

OBLIQUE-ANGLED SPHERICAL TRIGONOMETRY. 

The preceding investigations have had reference to right- 
angled spherical trigonometry only, but the application of 
these principles covers oblique-angled trigonometry also; for, 
every oblique-angled spherical triangle may be considered as 
made up of the sum or difference of two right-angled spheri¬ 
cal triangles. With this explanatory remark, we give 


TRIGONOMETRY. 


131 


\ 

I r ‘ * ' 

PROPOSITION I. 

In all spherical triangles , the sines of the sides are to each 
other as the sines of the angles opposite to them. 

This was proved in relation to right-angled triangles in 
Prop. III., Sec. 3, and we now apply the principle to oblique- 
angled triangles. 

Let ABC be the triangle, and let 
CD be perpendicular to AB , or to 
AB produced. 

Then, by Prop. Ill, R. A. Spher, Trig., 
we have, 

B : sin. A C == sin. A : sin .CD. 

Also, 

sin .CB\ B — sin .CD : sin.7?. 

By multiplying these two proportions together, term by 
term, and omitting the common factor B in the first couplet, 
and the common factor, si w.CD, in the second, w r e have 

sin .CB : sin.A£7 = sin.A : sin.i?. 


c 




PROPOSITION II. 

In any spherical triangle , if an arc of a great circle be let 
fall from any angle perpendicular to the opposite side as a 
base , or to the base produced , the cosines of the other two 
sides will be to each other as the cosines of the segments of the 
base. 

By the application of equation 8, (R. A. Spher. Trig.), to 
the last figure, we have, 

B cos .AC = cos.AZ* cos.DC 

B cos.BC = cos.DC cos. BD 


Similarly, 




132 


SURVEYING AND NAVIGATION. 


Dividing one of these equations by the other, omitting com 
mon factors in numerators and denominators, we have 

cos. AC 1 ' cos .AD 
cos.^6 7 — cos .BD 

Or, cos. AC: cos .BC = cos .AD : cos .BD. 


PROPOSITION III. 

If frcm any angle of a spherical triangle a perpendicular 
he let fall on the base, or on the base produced, the tangents of 
the segments of the base will be reciprocally proportional to 
the cotangents of the segments of the angle. 

By the application of Equation 2, (R. A. Spher. Trig.), to 
the last figure, we have, 

R sin .CD — tan .AD cot. A CD. 

Similarly, R svo.CD = tan .BD cot .BCD. 

Therefore, by equality, 

tan .AD cot.ACD = tan .BD cot .BCD 

Or, tan.AZ* : t&w.BD = cot .BCD : cot.ACD. 


PROPOSITION IV. 

The same construction remaining, the cosines of the angles 
at the extremities of the segments of the base are to each other 
as the sines of the segments of the opposite angle. 

Equation 7, (R. A. Spher. Trig.), applied to the triangle 
ACD, gives 

R cos. A = cos .CD sin. A CD (s) 

Also, R cos .B = cos.CD sm.BCD ( t ) 




TRIGONOMETRY. 


133 


Dividing equation (6') by (£), gives 


cos. A sin .ACD 
cos .B ~ sin .BCD 

Or, cos .B : cos. A = sin. BCD : sm.ACD. 


PROPOSITION Y. 

The same construction remaining , the sines of the segments 
of the base are to each other as the cotangents of the adjacent 
angles. 

Equation 1, (R. A. Spher. Trig.), applied to the triangle 
ACD , gives 

JR sin.AZ) = tan. CD cot.A ( s ) 

Similarly, JR sin .JBD = tan. CD cot.^ (i t ) 

Dividing ( s ) by ( t ), gives 

sin.AZ* cot.A 
si n.JBD ~ cot.i? 

Or, sin .BD : sm.AD = cot.B : cot.A. 


PROPOSITION YI. 

The same construction remaining , the cotangents of the two 
sides are to each other as the cosines of the segments of the 
angle. 

Equation 9, (R. A. Spher. Trig.), applied to the triangle 
ACD , gives 


Similarly, 


JR cos. A CD = cot. A C tan. CD 
JR cos.BCD = cot.. BC tan. CD 


( s ) 

V) 






134 


SURVEYING AND NAVIGATION. 


Dividing ( s ) by (t), gives 

cos. .4 CD cot .A C 
cos. BCD ~~ cot .DC 

Or, cot.AC: cot .BC — cos.ACD : cos.BCD. 


PROPOSITION VII. 

The cosine of any side of a spherical triangle is equal to 
the product of the cosines of the other two sides , plus the 
product of the sines of those sides multivlied by the cosine of 
the included angle. 

Let ABC be a spherical triangle, 
and CD a perpendicular from the 
angle C to the side AB , or to 
the side AB produced. Then, by 
Prop. II., 

cos.ylC: cos.CB = cos .AD : cos .BD (1) 

When CD falls within the triangle, 

BD = (. AB-AD ); 
and when CD falls without the triangle, 

BD = (AD—AB). 

Hence, cos .BD = cos .(AD — AB). 

How, cos .(AB—AD) = cos .(AD—AB), 

because each of them is equal to 

cos .AB cos.AD -\-sm.AB sin.HZ*, (Eq. 10, Prop. II., Plane 
Trig.). 

This value of cos .BD, put in proportion (1), gives 

cos.AC : cos .CB = cos.di) : cos.AB cos.ylZ) -fsin..dZ? sin.-di) (2) 

Dividing the last couplet of proportion (2) by cos.HZ*, 
observing that 

sin.HZ) 

cXXD = tan -^ 2> » 










TRIGONOMETRY. 


135 


and we have 

cos .AC: cos. CB = 1 : cos.Ai? +sin.Ai? tan.AD (3) 

By applying equation 6, (R. A. Spher. Trig.), to the triangle 
ACD, taking the radius as unity, we have 

cos. A = cot .AC tan .AD {Jc) 

But, tan.A6 y cot.A(7 = 1, (Eq. 5., Plane Trig.) ( l ) 

Multiply equation ( k ) by tan .A C, observing equation (7), 
and we have 

* 

tan.Af7cos.A = tan .AD. 

Substituting this value of tan.AD in proportion (3), we 
have 

cos. A 77: cos .CB = 1 : cos.AD + sin.AD tan.AC'cos.A (4) 

Multiplying extremes and means, gives 
cos .CB = cos.Af7 cos.AD-bsin.ADicos.Af/ tan .AG) cos. A. 

But, tan.AD — or, cos .AC tan .AC = sin. AC. 

cos. A C 

Therefore, 

cos .CB = cos.AC cos.AD + sin.AD sin. A 77 cos.A. 

If the sides opposite the angles, A, B , and C, be respectively 
represented by a , #, and <?, this equation becomes, 

cos.a = cos.b cos.c-f-sin.J sin.c cos.A. 

This formula conforms to the enunciation in respect to the 
side a. Interchanging b and a and writing B for A, in the 
last equation, we get the formula for cos .b, which is, 

cos .b = cos .a cos.tf+sin.a sin.c cos.D. 

Interchanging c and a and writing C for A, we get the 
formula for cos.c, which is, 

cos.c = cos .a cos.£ + sin.& sin.J cos.D 

Hence, we have the three symmetrical formulae: 

cos .a — cos .b cos.c-fsin.5 sin.c cos. A 
cos .b = cos.a eos.c + sin.a sin.c cos .B 

m 

cos.c = cos .a cos.J + sin.« sin.5 cos. G 


(S) 



136 SURVEYING AND NAVIGATION. 


From these, by simple transposition and division, we deduce 
the following formulae for the cosines of the angles of any 
spherical triangle, viz.: 


cos.a—cos.5 cos.c " 

cos.^4 =-•— 7 —•- 

sin .b sin.c 

cos.J — cos.a cos.c 

COS.Yj = -:-.- 

sm .a sin. 6 1 
cos.c—cos.d cos.J 

COS. 6 = -7-7-7- 

sin.& sm .o 

By means of these equations we can find the cosine of any 
of the three angles of a spherical triangle in terms of the 
functions of the sides; but in their present form they are not 
suited for the employment of logarithms, and we should be 
compelled to use a table of natural sines and cosines, and to 
perform tedious numerical operations, to obtain the value of 
the angle. 

They are, however, by the following process, transformed 
into others well adapted to the use of logarithms. 

In Eq. 34, Plane Trig., we have 



1-f cos.^4 = 2cos. 2 \A. 

Therefore, 2 cos.^.4 = 1 + cos.a-cos.^ oos^, 

sin .b sin.c 

2 cos 2 i A = sin.c—cos.5 cos.c) + cos.a ,, 

sin .b sin .c K *' 


But by Equation 9, Plane Trigonometry, 

cos. (J + c) = cos 1) cos .c —sin .b sin.c, 

Or, sin.J sin.c—cos.J cos.c = —cos. (& + c). 

Substituting for the first member of this equation, as found 
in (Eq. m), its value,—cos. (& + <?), we have 

2 cos . 2 ±A — vo ^ a ~ C QS - (?> + c) 

bin.b sin.c 










» 


TRIGONOMETRY. 


137 


Considering (b + c) as one arc, and then making application 
of equation (18), Plane Trigonometry, we have, 


2 sin 


2 cos. * 2 = 


• /(i~\-b-\-c\ • (b -J- c — o\ 

m. ( ) sm.| - j 


sin.6 sin.c 

But, - i ~ c ~ - = a; and if we put S to represent 


2 


£i^, we shall have, 
2 


cos. 


,^1 _ sin./S' sin. (/S'—#) 


2 


sin.& sin.c 


Or, 


A _ . /sin .S sin. (/S— a) 

IT — r • T—•-• 

2 sm.6 sm.c 


The second member of this equation gives the value of the 
cosine when the radius is unity. To a greater radius, the 
cosine would be greater; and in just the same proportion as 
the radius increases, all the trigonometrical lines increase; 
therefore, to adapt the above equation to our tables where the 
radius is R , we must write R in the second member, as a fac¬ 
tor ; and if we put it under the radical sign, we must write R a . 

For the other angles, we shall have precisely similar equa¬ 
tions. 



To deduce from formulae (/S’), formulae for the sines of the 
half of each of the angles of a spherical triangle, we proceed as 
follows: 

From Eq. 35, Plane Trig., w T e have 

2 sin. 3 ^ = 1—cos.A. 





















138 


SURVEYING AND NAVIGATION. 


Substituting the value of cos.A, taken from formulae ( S ), 
and we have, 

cos.«—cos .b cos.c 


2 sin. 2 lA = 1 — 


sin. 6 sin.c 
(sin.Z> sin.<?-f-cos.& cos.c)—cos.« 


(o) 


sin.£ sin.c 

But, cos. (b^c) = sin.& sin.c-f cos.& cos.tf, (Eq. 10, Plane 
Trig.) 

This equation reduces equation ( o ) to 
2 sin .*U = 

sin.£ sin.c 

Considering ( b^c ) as a single arc, and applying equation 
18, Plane Trig., we have 

<a-\-b — c\ niw% (a + c—by 

' («0 


2 sin 


2 sin. HA = 


ia-\-b—c\ . (a + c — b\ 

m * ( 2 ) bin * ( 2 / 


sin.J sin.c 


t> , CL~\~b — G d~\~b~\~C rr • . nr 

But. —— - = ———— — c = o — Cy it we put o — 


2 


2 


® -f- 5 “I - G 

2 ? IS' j 

Also, = £-J. 

’ 2 2 

Dividing equation (o') by 2, and making these substitutions, 
we have, 

. . sin. (S—c) sin. (S—b) 

sm. 2 ±A = - v -- 7 , . 2 


sin.J sin.c 


when radius is unity. 

When radius is i?, we have 


8 i n i A = J /l ' 2silK ( ,y ~ c ) 8in - ( S ~ h ) 

sin.£> sin.c 

Similarly, sin.ii? = 


sm.a sin.c 


g in i (j — J R 2 $\n\S—d) sin .(S—b) 

sin.a sin.6 


(ff) 


And, 
























139 


TRIGONOMETRY. 


The preceding equations are now adapted to our tables. 
We shall show the application of these formulae, and those in 
group (T 7 ), hereafter. 


PROPOSITION VIII. 


The cosine of any of the angles of a spherical triangle , is 
equal to the product of the sines of the other two angles multi¬ 
plied by the cosine of the included side , minus the p>roduct of 
the cosines of these other two angles . 


Let ABC be a spherical triangle, and C' 

A'B'C its supplemental or polar triangle, 
the angles of the first being denoted by A , 

B , and 0, and the sides opposite these 
angles by a , b , c , respectively; A\ B ', CK 
a\ b\ c\ denoting the angles and corres¬ 
ponding sides of the second. 

By Prop. VI., Spherical Geometry, we have the follow¬ 
ing relations between the sides and angles of these two trian¬ 
gles. 



A' = 180 °—a, B’ = 180 °—b, C = 180°—<?; 
a* = 180 °-A, V = 180°-i?, d = 180°-f7. 


The first of formulae (V), Prop. VII., when applied to the 
polar triangle, gives 

cos.&' = cos.J' cos.<f-t-sin.^' sin.e' cos.Eb; (1) 

which, by substituting the values of a\ If c\ and A\ becomes 

cos. (180°— A) = cos.(180°-^) cos. (180°-G) +sin. (180°- 

B) sin. (180°— C) cos. (180°— a). (2) 

But, 


cos. (180°— A) — —cos.J., etc.; sin. (180°— B) = sin.i?, etc.; 
and placing these values for their equals in eq. (2), and chang- 





140 






SUKVEYING AND NAVIGATION. 

ing the signs of both members of the resulting equation, we 
get 

cos.H = sin.j5 sin. (7 cos.a—cos.^ cos. (7, 

which agrees with the enunciation. 

By treating the other two of formulae ($), Prop. VII., in the 
same manner, we should obtain similar values for the cosines 
of the other two angles of the triangle ABC / or we may get 
them more easily by a simple permutation of the letters A , B , 
C ’ a , etc. 

Hence, we have the three equations 
cos.A 

COS.J5 
cos. £7 


= sin.2? sin. (7 cos.«— cos.jS cos. (7 ) 

= sin.A sin .C cos .b —cos. A cos .C (V) 
= sin. A sin.i? cos.c—cos. A cos .B ' 


By transposition and division, these equations become 

cos. A + cos.7? cos. C 


cos.« 


cos.J = 


cos.<? = 


sin../? sin. C 

cos._Z?-f-cos.A cos. C 
sin. A sin.6 7 
cos. C-\- cos. A cos .B 
sin.A sin.^? 


( 3 ) 


From these we can find formulae to express the sine oi the 
cosine of one half of the side of a spherical triangle, in terms 
of the functions of its angles ; thus : 

Add 1 to each member of eq. (3), and we have 


1-f COS.& = 


cos.A-fcos.7? cos. (7-4-sin.A? sin.(7 _ 
sin.i> sin. 6 

cos. A +cos.(i?— C) 
si n.B sin .C 


1-fcos.a = 2 cos. 2 i&; hence, 

2 cos. Ha - cos.^4 + cos.(i?— C) 

sin .B sin.C' 


But, 














TRIGONOMETRY. 


141 


and since cos.^L-fcos. (B—C) = 2 cos.|(J. + J5— 67) cos4(J. 
+ C — B)^ (Eq. 17, Plane Trig.), we have 

2 cos. 2 \cl = 2 cos.i(^l + B- C) cos .{(A + C— B) 

sin.i? sin. G 

Make A + B+C — 2S-, then A+B-C= ZS—ZC,A + C 
—B — 2S—2-B, l(A + £—C) = S — C, and %(A + C—B =s 
ft—B ; whence 



sin..# sin. 0 



or, 

W'^p-oyxm.is-B) 



* sm.Yjsm.6 



Similarly, 

cos# _ y^-A)^S-G) 

2 sin.^1 sm.6 

*■ 

(F) 

and, 

, ./cos.(S--A)co&.(jS—B) 



cos. 2 tf _ y sill gin B 

J. 



To find the 

sin.la in terms of the functions of the 

angles. 


we must subtract each member of eq. (3) from 1, by which 
we get 


1—COS.& 


-j __cos.^4 -f cos.i? cos. 67 
sin .B sin. 67 


Hut, 1—cos.& = 2sin. 2 2& ; hence we have, 

q . 21 _ (sin.^sin. 67— cos.Bcos.C) —cos.^4 

■“Sin. ---; V,—. -• 

sm.Yj sm. 6 

Operating upon this in a manner analogous to that by which 
cos ,\a was found, we get, 


sin.^ = 
sin.= 
ein.^<? = 


—cos./# cos. (£— A) ) ^ 
sin.i? sin. 67 f 
— COS.# COS .(#— B) ) 2 
sin .A sin.6 r I 
—cos.# cos. (#—67) ) \ 
sin.^4 sin.i? i 


(W) 
























142 SURVEYING AND NAVIGATION. 

If the first equation in (TT) be divided by tlie first in ( T r/ ), 
we shall have, 



And corresponding expressions may be obtained for tan.^3 
and tan.^c. 

SOLUTION OF OBLIQUE-ANGLED SPHERICAL TRIANGLES. 

All cases of oblique-angled spherical trigonometry may be 
solved by right-angled Trigonometry, except two; because 
every oblique-angled spherical triangle is composed of the 
Sum, or the difference, of two right-angled spherical triangles. 

When a side and two of the angles , or an angle and two of 
the sides are given , to find the other parts, conform to the fol¬ 
lowing directions: 

Let a perpendicular be drawn from an extremity of a given 
side, and opposite a given angle or its supplement; this will 
form two right-angled spherical triangles; and one of them 
will have its hypotenuse and one of its adjacent angles given, 
from which all its other parts can be computed ; and some of 
these parts will become as known parts to the other triangle, 
from which all its parts can be computed. 

To facilitate these computations, we here give a summary 
of the practical truths demonstrated in the foregoing proposi¬ 
tions. 

1. The sines of the sides of spherical triangles are propor¬ 
tional to the sines of their opposite angles. 

2. The sines of the segments of the bases, made by a perpen¬ 
dicular from the opposite angle , are proportional to the cotan- 
gents of their adjacent angles. 

3. The cosines of the segments of the base are proportional 
to the cosines of the adjacent sides of the triangle. 

4. The tangents of the segments of the base are reciprocally 



TRIGONOMETRY. 


143 


proportional to the cotangents of the segments of the vertical 
angle. 

5. The oosines of the angles at the base are proportional to 

the sines of the corresponding segments of the vertical angle. 

* 

6. The cosines of the segments of the vertical angle are pro¬ 
portional to the cotangents of the adjoining sides of the tri¬ 
angle. 

The two cases in which right-angled spherical triangles are 
not used, are, 

1st. When the three sides are given to find the angles ; and, 

2d. When the three angles are given to find the sides. 

The first of these cases is the most important of all, and for 
that reason great attention has been given to it, and two series 
of equations, (Tand £7, Prop. VII.), have been deduced to 
facilitate its solution. 

As heretofore, let ABC represent any triangle whose 
angles are denoted by A, B, and C\ and sides by #, b , and c ; 
tne side a being opposite L A , the side b opposite L B , etc. 


EXAMPLES. 




1. In the triangle ABC , a = 70° 4' 18"; b = 63° 21’ 27" ; 
and c, 59° 16* 23"; required the angle A. 

The formula for this is the first equation in group ( T ' Prop. 
VII.), which is 


A _ [B 2 sin.,#sin.(A— a)\£ 


cos.— = 


2 \ sin. b. sin.c 


Y 


We write the second member of this equation thus: 


\/1 ) (sm./S)sm.(S— a), 

7 \sm.b) \sin.c/ 

showing four distinct factors under the radical. 

The logarithm corresponding to is that of sin A sub 








144 


SURVEYING AND NAVIGATION. 


D 

traded from 10 ; and of _— is that of sin.c subtracted from 

Slll.tf 

10, which we call sin.complement. 

BC = a = 70° 4' 18” 

AB = c = 59° 16' 23” sin.com. .065697 

AC = b = 63° 21' 27” sin.com. .048749 

2 )192° 42' 8” 

S = 96° 21' 4” sin. 9.997326 

S-a = 26° 16' 46” sin. 9.646158 

2)19.757930 

\A = 40° 49' 10” cos. 9.878965 

_ 2 _ 

A = 81° 38' 20” 


When we apply the equation to find the angle A , we write 
a first at the top of the column: when we apply the equation 
to find the angle B , we write b at the top of the column. 
Thus, 

To find the angle B. 


./B 2 sin./S' sin .(S^b) 
cos J3 — f • ~ • 


sin.a sm .c 


=V(~) (A-) (sin .S) sin.(/S’— V) 
'sin.a/ 'sm. 

l = C3° 21' 27" 

c — 59° 10' 2S" sin.com. .065697 

a = 70° 4' 18" sin.com. .026875 

2) 192° 42' 8" 

S = 96° 21' 4" sin. 9.997326 

£—l = 32° 59' 37'' sin. 9.736034 

2)19.825872 

\B = 35° 4' 49" cos. 9.912936 

2 

B = 70° 9' 38" 


















145 


TRIGONOMETRY. 


By the other equation in formulae (T 7 , Prop. YIL), we can 
find the angle C ; but, for the sake of variety, we will find the 
angle C by the application of the third equation in formulae 
(Z7, Prop. VIL). 


sin 


^ -/J? 2 sin.(>S r -— ft) sin.($— a) 

" 2 sin.J sin.tf 


= |/ &m.(S—b) sin.OS— a) 
\sin .a! 


c = 59° 16' 23” 

a — 70° 4' 18” sin. com. .026817 
ft = 63° 21 7 27” sin. com. .048479 


8 = 

8—a = 

8-ft = 
W = 


2)192° 42 f 8” 

"96° 21' 4” 

26° 1& 46” sin. 
32° 59' 37” sin. 


32° 23' 17” sin. 
2 


9.646158 

9.736034 

2) 19.457488 

9.778744 


C = 


64° 46' 34” 

To show the harmony and practical utility of these two sets 
of equations, we will find the angle A from the equation 


sin.i^l 


i/ (—Lf) (-^—)sin.(8—ft) sin .(8—c). 

\sin .ft! Vsin.c/ 


a = 70° 4' 18” 

ft — 63° 21' 27” sin.com. 

c = 59° 16' 23” sin.com. 

2 ) 192° 42^ 8” 


.048749 

.065697 


8 = 96° 21' 4” 

8-ft = 32° 59' 37” sin. 
8—c ’== 37° 4' 41” sin. 


•A = 40° 49' 10” sin. 


9.736034 

9.780247 

2) 19.630727 

9.815363 


2 


A = 81° 38' 20” 




















146 


SURVEYING AND NAVIGATION. 


2. In a spherical triangle ABC, given the angle A , 38° 19' 
18" ; the angle B, 48° O' 10"; and the angle C 121° 8 f 6 "; to 
find the sides a, &, £. 

By passing to the triangle polar to this, we have, (Prop. 
VI., Spherical Geometry), 

A = 38° 19' 18''; supplement 141° 40' 42" 

B — 48° 0 f 10"; supplement 131° 59' 50" 

C = 121° 8' 6"; supplement 58° 51 ; 54" 

We now find the angles to the spherical triangle, the sides 


of which 

are these supplements. 



Thus, 

141° 40' 42“ 
131° 59' 50" 
58° 51' 54” 

% 

sin.com. .128909 

sin.com. .067551 


2) 332° 32' 26" 

• 



166° 16' 13" 
24° 35' 31" 

sin. 

sin. 

9.375375 

9.619253 




2)19.191088 


66° 47' 371" 
2 

/ 

cos. 

9.595544 

Angle = 133° 35' 15" 

Supplement = 40° 24' 45" 




Therefore a , of original triangle = 46° 24 f 45" In the 
same manner, we find b = 60° 14' 25", and c = 89 G l f 14". 

It is perhaps better to avoid this indirect process of com¬ 
puting the sides of a spherical triangle when the angles are 
given, by the application of the equations in group V or W , 
Prop. VIII., O. A. Spher. Trig. We will illustrate their use 
by applying the second equation in group (IT), for computing 
the side b. This equation is 








' 

TRIGONOMETRY. 147 

6 i n ij _ / —cos.xg' cos.QS'—i?) U 

\ bin.^1 bin. 6' / 

A = 38° 19' 18" 

B = 48° O' 10" 

C - 121° 8' 6" 

2)207° 27' 34" 

8 = 103° 43' 47"—cos.£=+sin.l3°43 l 47"=9.375376 
B = 48° O' 10" cos.(S-B)^55°43' 37" = 9.750612 

(8—B) = 55° 43' 37" 2 ) 19.125988 

square root = 9.562994 
sin.A = 38° 19' 18" = 9.792445 
sin. C = 121° 8' 6" = 9.932443 

2 ) 19.724888 

square root = 9.862444 = 9.862444 

diff. -1.700550 

Add 10, for radius of the table, 10 

Tabular sin .\b = 30° 7' 14" = 9.700550 

2 

b = 60° 14' 28", nearly. 


PRACTICAL PROBLEMS. 

1. In any triangle, ABC , whose sides are a, b, e , given 
b _ ngo 2 ' 14", c =v 120° 18' 33", and the included angle 
A — 27° 22' 34", to find the other parts. 

( a = 23° 57' 13", angle i? = 91° 26' 44", and 
A71S ‘ I G = 102° 5' 52". . 

2. Given, A = 81° 38' 17", B - 70° 9' 38", and C = 64* 

46' 32", to find the sides b , c. 

' a - 70° 4' 13", b — 63° 21' 24", and 




c = 59° 16' 21". 















148 


SURVEYING AND NAVIGATION. 


3. Given, the three sides, a = 93° 27' 34", b = 100° 4 1 26", 
and c = 96° 14 ; 50", to find the angles A, B , and C. 

A = 94° 39' 4", B = 100° 32' 19", and G = 96° 



58' 35". 


4. Given, two sides, 5 = 84° 16', c = 81° 12 f , and the angle 
C = 80° 28', to find the other parts. 

"The result is ambiguous, for we may consider the 
angle B as acute or obtuse. If the angle B is 
. acute, then A = 97° 13' 45', B = 83° 11' 24", 

and a = 96° 13' 33". If B is obtuse, then A = 
21° 16' 43", B = 96° 48' 36", and a = 21° 19' 


29". 


5. Given, one side, c = 64° 26', and the angles adjacent, A 
= 49°, and B — 52°, to find the other parts. 

Ans \b =45° 56' 46", a = 43° 29' 49", and 
t C = 98° 28' 4". 

6. Given, the three sides, a = 90°, b = 90°, v = 90°, to find 
r he angles A, B , and C. 

Ans. A = 90°, B = 90°,• and C = 90°. 

7. Given, the two sides, a = 77° 25' 11", c = 128° 13' 47", 
and the angle C = 131° 11' 12", to find the other parts. 

A ( b = 84° 29' 20", A = 69° 13' 59", and 
1 B = 72° 28- 42". 

8. Given, the three sides, a = 68° 34 1 13", b = 59° 21' 18’, 
and c — 112° 16' 32”, to find the angles A, £, and C. 


A = 45° 26' 38", B = 41° 11' 30" 
C = 134° 53' 55”. 



9. Given, a = 89° 21' 37'', i = 97° 18' 39", c — 86° 53' 
46", to find G, 77, and (7. 

^ I = 88° 57' 20", 5 = 97° 21' 26", 

' ( C = 86° 47' I?". 

P * 




TRIGONOMETRY. 


149 


10. Given, a — 31° 26' 41", c = 43° 22' 13", and the angle 
A = 12° 16', to find the other parts. 

f Ambiguous; b = 73° 7' 34", or 12° 17' 40"; angle 

Ana. -J B = 157° 3' 44", or 4° 5S' 30" ; 0 = 16° 14’ 27", 
( or 163° 45' 33". 

11. In a triangle, ABC , we have the angle A = 56° 18f 
40", B = 39° 10' 38"; AD , one of the segments of the base, 
is 32° 54' 16". The point D falls upon the base AB , and the 
angle C is obtuse. Required the sides of the triangle and the 
angle C. 

f Ambiguous ; C = 135° 25 f , or 135° 57'; c — 122° 

Ans. -j 29', or 123° 19 f ; a = 89° 40', or 90° 20'; b = 49° 
( 23' 41". 

12. Given, A = 80° 10' 10", B = 58° 48' 36", C= 91° 52' 
42", to find a , 5, and c. 

Ans. a = 79° 38' 21", b = 58° 39' 16", c = 86° 12' 52". 


CHAPTER III. 


OF MENSURATION. 

Mensuration is the measurement of extension. 

Measurement is the application of a certain fixed portion 
of extension to that which is measured; this portion is called 
a unit. 

The measure of anything—or the number expressing the 
result of measurement—is the ratio between the unit and 
that which is measured. 

The quality or dimension of the unit corresponds to that 
which is measured; line is applied to line, surface to surface, 
solid to solid. 

All ratios are abstract, the idea of quality disappearing in 
the division. Thus, 

4 feet are contained in 12 feet, 3 times ; or the ratio between 
them is 3. 

4 yards are contained in 12 yards , 3 times; or the ratio 
between them is also 3. 

Hence, though measurement is strictly an application of a 
unit of the same kind, yet, if we can prove the ratio between 
surfaces or solids to be the same as the ratio between lines, we 
m av, by a comparison or measurement of lines, determine the 
measurement of surface and solidity. We shall find this fre¬ 
quently in use. 

Mensuration is naturally divided into the measurement of 
lines, surfaces, and solids. 

We shall call attention to these successively. 




MENSURATION. 


151 


SECTION I. 

OF LINES. 

In the measurement of those lines which are accessible, 
some convenient unit of measure is directly applied. j 

All measurement of lines is called linear measurement, and 
units which are lines are linear units. 

The following Tables show the standard linear units in this 
country. 


TABLE OF LONG MEASURE. 


12 inches (in.) make 1 foot,.ft. 

3 feet make 1 yard, . . ...yd. 

5* yards, or 16£ feet, make 1 rod,.rd. 

40 rods make 1 furlong,.fur. 


8 furlongs, or 320 rods, make 1 statute mile, . . . mi. 

The following are also in use : 


6 feet make 

1.15 statute miles make 

3 geographic miles make 

60 geographic miles or make 

69.16 statute miles 


1 fathom, used in measuring depths at sea. 

( used in measuring dis- 
1 geographic mile, -< ® 

( tances at sea. 

1 league. 

, (of latitude on a meridian, or of 
1 degree < ’ 

t longitude on the equator. 


TABLE OF CIRCULAR MEASURE. 


60 seconds (") 
60 minutes 
360 degrees 


make 1 minute (') 

make 1 degree ( ° ) 

make 1 circumference of circle (circ.) 


surveyors’ long measure. 

7.92 inches (in.) make 1 link,.1. 

25 links make 1 rod,.rd. 

4 rods, or 66 feet, make 1 chain,.... ch. 

80 chains make 1 mile,.mi. 


When lines are inaccessible, the principles of Geometry and 
Trigonometry are used to derive from accessible or known con¬ 
ditions, lines otherwise unknown This application is called 









SURVEYING- AND NAVIGATION. 


152 


MENSURATION OF HEIGHTS AND DISTANCES 

DEFINITIONS.. 


A Vertical Line is one formed by a line and plummet freely 
suspended and allowed to come to rest. Such a line is per¬ 
pendicular to a tangent to the earth’s surface, and is a contin¬ 
uation of a radius of the earth. 


A Vertical Plane is one passing through a vertical line. 

A Horizontal Line is one at right angles to the vertical line. 
A Horizontal Plane is one perpendicular to the vertical line. 

Angles are vertical or horizontal, as the planes of their sides 
are vertical or horizontal. 


An Angle of Elevation is one r the plane of whose sides is ver¬ 
tical, and where one side is horizontal and the other ascending. 

An Angle of Depression lies also in a vertical plane, but with 
one of its sides horizontal, and the 
other descending. 

For example, in the figure fol¬ 
lowing, EBB is an angle of ele¬ 
vation, having the ascending line 
BB ; EBD is an angle of depres¬ 
sion, having BE horizontal and 
BD descending- 

Stations are points, selected for D 
convenience or at pleasure, from 
which angles and lines are mea¬ 
sured. 

A Base Line is one measured as a basis or known quantity, 
to be used in calculations; and with this the lines and angles 
to be determined are connected. 



PROBLEM I. 

To determine the vertical height oj^ an object above a hori¬ 
zontal plane. 









MENSURATION. 


153 


Case 1. When the object is vertical, and its base accessible. 

From the base of the object, measure on the horizontal 
plane a line of convenient length, and at its extremity take 
the angle of elevation. 

Then if AB is the object and B 

AC the base line, the instrument 
being placed at CD, there will be 
known in the triangle DEB, 
base DE — AC, and base angle 
EDB = angle of elevation. 

By (Prop. Ill, Plane Trigonome¬ 
try, Chap. 2.) 

B : DE — ta w.EDB : BE. 

Whence BE is known; EA — 

CD — height of instrument for 
measuring angles, BE-\-EA = AB, vertical height required. 

EXAMPLES. 

1. Required the height of an object when the line measured 
from base to station is 110 feet, and angle of elevation is found 
to be 54° 32', height of instrument being 5 feet. 

Solution. 

In last figure AC = DE — 110 feet, 

L EDB = 54° 32b 
R : 110 : : tan. 54° 32' : BE. 

By logarithms, log. 110 = 2.041393 

Log. tan. 54° 32' = 10147267 

• 12.188660 

Log. R =10 

2.188660 = log.154.4 = BE. 

By condition EA — 5 feet: hence BE-\-EA = 154.4 -f 5 
= 159.4 feet, height of object. 















154 


SURVEYING AND NAVIGATION, 


2. The distance from foot of a tower to station is found to 
be 109 feet, and angle of elevation 39° 22', required the height 
of tower, allowing 5 feet for height of instrument. 

Ans. 94.427 feet. 

3. Given, base line = 100 feet, 

Angle of elevation = 47° 52', 

Allowing 5 feet for instrument, required height of object. 

Ans. 115.54 feet. 

4. Given, base line = 80 feet, 

Angle of elevation = 32° 42 f , 

Allowing for instrument as before, required height of object. 


Case 2.—When the object, or its base, is wholly inaccessible, 
two methods may be employed. 


1st Method. —Two stations having been taken in a direct 
line towards the object, measure the distance between these 
stations as a base line; also, the angle of elevation of object 
at each station. A 


Then, in figure, in the triangle ABC ', 
there will be known, BC the base line, the 
angle ACB , angle of elevation at first sta¬ 
tion, and angle ABC — 180°— ABB, angle 
of elevation at second station. 

Angle CAB = (ABB-ACB) 



By Plane Trig., Chap. 2d, Sec. 2, Prop. IY, 


sin.CAB : BC = sin.A CB : AB. (2) 


Then in the right-angled triangle ABB , there are known 
AB from (2), and ABB. 

By Plane Trig., Prop. Ill, 

R : AB = sin .ABB : AB. (3) 


To AB add height of instrument as in Case 1, and we ob¬ 
tain vertical height of object. 




MENSURATION. 


155 


EXAMPLES. 

1. Wishing to know the vertical height of an object upon a 
hillside, above the horizontal plane, I took a station in the 
plain below the hill, at which the angle of elevation of the 
object was 30° 48'. I then measured toward the object 100 
feet to a second station on a level with the first, where I found 
the angle of elevation 44° 28'. Required vertical height above 
the plane of stations, allowing 5 feet for instrument. 


Solution. 


In last figure, BC = 100 feet, [_ ACB = 30° 48', L ABD 
44° 28'. 

L ABC = 180°—44° 28' = 135° 32' 

L CAB = (44° 28'—30° 48') = 13° 40'. (1) 

sin. 13° 40' : 100 = sin. 30° 48' : AB. (2) 

Co. log. sin. 13° 40' = 0.626586 
Log. 100 = 2.000000 

Log. sin. 30° 48' = 9.109306 

Sum less 10, = 2.335892 = log. 216.72. 

AB = 216.72. 

B : 216.72 = sin. 44° 28' : AB. (3) 

Log. 216.72 - 2.335892 

Log. sin. 44° 28' = 9.845405 

Less 10 = 2.181297 = log. AD. 

2.181297 = log. 151.81 151.81 = AD 

5 

Ana. = 156.81 


2. Wishing to ascertain the height of a church spire, I meas¬ 
ure near its foot an angle of elevation o3° 45'; then Preced¬ 
ing 120 feet in a line with first station and spire, I find the 
angle of elevation only 34°. What is the height of spire, 
allowing as before for instrument. Ana. 165.14 





156 


SUKVEYING AND NAVIGATION. 


3. Given, a*ngle of elevation first station = 60° 12'. 

Angle “ second “ = 42° 25 ; . 

CJ 

Distance between stations = 72 feet. 


Required height of object, allowing 5 feet for instrument. 

Ans. 142.98 feet. 


2d Method. —Select two stations, from which both base and 
top of the object are visible, and measure distance between 
these for a base line. At each station measure horizontal angle 
between foot of object and the other station ; also, at one sta¬ 
tion measure the angle of elevation of the top of the object. 

Then, if CD be the object, and A and B 
the assumed stations, in the horizontal trian¬ 
gle ABC\ there will be known AB , the 
base line, and the horizontal angles ABC 
and BA C. 


D 



Also, ACB = 180°— (ABC+ BAC). 

By Prop. IY, Sec. 2, Chap. 2, 

sin .ACB : AB = sin .ABC: AC , or 

= sin .BAC : BC. 


(i) 


( 2 ) 


Then, in the right-angled triangle A CD (or BCD , if the 
angle of elevation be taken at B), 

(Prop. Ill, Sec. 2, Chap. 2), 

R : AC — tan .CAD : CD, (3) 

Or, if BCD be used, R : BC — tzw.CBD : CD. 

To CD add height of instrument as before, and the sum 
will be the vertical height of the object. 

By an application of this method, we may compute the dif¬ 
ference of level between two 
horizontal planes, if the same 
object is visible from both. 

For example, let M be a pro¬ 
minent tree or rock near the 
top of a mountain ; by observa¬ 
tions taken at A, we can deter- 


M 









MENSURATION. 


157 


mine the perpendicular Mr. By like observations taken at 
B , we can determine the perpendicular Mm. The difference 
between these two perpendiculars is n?n, or the difference in 
the elevation between the two points A and B. If the dis¬ 
tances between A and n, or B and ra, are considerable, or 
more than two or three miles, corrections must be made for 
the convexity of the earth; but for less distances, such cor¬ 
rections are not necessary. 

EXAMPLES. 

1. It is desired to determine the vertical height of a light¬ 
house near the mouth of a harbor. Two stations are selected 
on the shore, distant from each other 160 feet. The horizon¬ 
tal angle at the first station is found to be 88° 44', and the 
angle of elevation at the same point 25° 56' ; the horizontal 
angle at the second station is found to be 60° 37'. Required 
height of light-house. 

Solution. 

In preceding figure, if CD be the light-house, A and B the 
first and second stations, there are known AB = 160, L CAB 
= 88° 44', and L ABC = 60° 37'. 

Also, ACB = 180°-(88° 44'+ 60° 37') = 30° 39'. (1) 

sin. 30° 39' : 160 = sin. 60° 37' : AC. (2) 

Co. log. sin. 30° 39' = 0.292607 

Log. 160 = 2.204120 

Log. sin. 60° 37' = 9.940196 

Log. AC = 2.436923 = log. 273.48. 

In triangle ACD , [_ CAD — 25° 56'. 

R : 273.48 = tan. 25° 56' : CD. 

Log. 273.48 = 2.436923 

Log. tan. 25° 56' = 9.686898 

Less 10 = 2.123821 = log. CD. 

2.123821 = log. 133 nearly. 133-1-5 = 13b feet. An8. 




158 


SURVEYING AND NAVIGATION. 


2. The elevation of the top of a tower at one station is 19° 45' 
10", and the horizontal angle at the same station between foot 
of tower and a second station is 91° 40'. At the second sta¬ 
tion, distant from first 400 feet, the horizontal angle is 51° 50'. 
Required height of tower. 

Ans. 194.85 feet. 

3. The angle of elevation of aspire from an assumed station 
is 23° 50' 17", and the horizontal angle at the same point, 
between the foot of the spire and a second station, distant 416 
feet, is 93° 4' 20". At the second station the horizontal angle 
is 54° 28' 36. Required height of spire. 

Ans. 278.8 feet. 


PROBLEM II 


The height of an object being known , to determine its dis¬ 
tance from a visible station in the horizontal plane at its foot. 

If the observer be at the station, let the a'ngle of elevation 
be measured; if on the object, the angle of depression, which 
equals that of elevation. B 

Then, if EB is the object, 
there will be known in the right- 
angled triangle DEB , one side 
BE \ and the angle EDB , either 
by measurement, or because 
FED = EDB. 

By Plane Trig., Chap. 2, Sec. B 
2, Prop. Ill, 

DE : BE — B : tan. EDB. 

Whence is known DE ' the required distance. 

X. B. If angle of elevation is taken, the height of the in¬ 
strument should be deducted from the vertical height of the 
object to obtain BE. 










MENSURATION. 


159 


EXAMPLES. 

1. Required the distance from a given station to a spire 170 

feet in height, the angle of elevation at the station being: 
25° 32'. 

Solution. 

In triangle BBE, BE = 170—5 = 165 feet, 

L EBB = 25° 32'. 

BE : 165 = R : tan. 25° 32 f . 

log. R + log. 165 s= 12.217484 
Subtract log. tan. 25° 32' = 9.679146 

Log. BE =. 2.538338 = log. 345.41. 

Ans. = 345.41 feet. 

2. Wishing to know rny distance from a building on the 
opposite side of a river, knowing the height of the building to 
be 52 feet, I measured its angle of elevation 10° 18'. What 
was my distance ? 

Ans. 258.62 feet. 

3. From the top of a mast of a vessel, 75 feet above the 
water, the angle of depression of another vessel’s hull was 
found to be 18° 30'. What was the distance between the 
vessels ? 

4. Being upon the top of a tower 92 feet high, I measured 
the angle of depression of the bottom of a building, 5° 48'; 
required the distance between tower and building. 


PROBLEM III. 

To determine the distance between two distant objects. 

Case 1.—When the objects, though separated by an impass¬ 
able barrier, are themselves accessible and visible from an 
assumed station. 



160 


SURVEYING AND NAVIGATION. 


Measure the distance from each 
object to the assumed station, and 
then take the horizontal angle at 
the station. 

If A and B represent the objects, 
rnd C the assumed station, there 
Mill be known AC\ BC and L C. 

By Plane Trig., Chap. 2, Sec. 2, Prop VTI. 

AC+BC: AC-BC = ta n.$(A+B) : tan .{(A~B) (1) 
i(A + £) + l(A-B) = A, 

or larger angle, opposite greater side. 

By Plane Trig. Chap. 2, Sec. 2, Prop. IV. 

Sin.M : BC — sin.6 r : AB, the required distance. (2) 

EXAMPLES. 

1. It is desired to determine the distance between two objects 
separated by M’oods and marsh. From the tw T c objects the dis¬ 
tances are measured to a convenient point, 122 and 161 yards; 
the horizontal angle at the station is found to be 52° 42'. 
What is the distance ? 



Solution. 

AC = 162, BC = 121, and \_ACB = 52° 42', 

A + B = 180°-52° 42' = 127° 18'; &A + B) = 63° 39'* 

283 : 39 = tan. 63° 39' : tan.£(M— B) (1) 

Co. log. 283 = 7.548214 

Log. 39 = 1.591065 

Log. tan. 63° 39' = 10.305117 

Sum less 10 = 9.444396 = log. tan .{{A—B) = 

log. tan. 15° 32' 52". 





MENSURATION. 


161 


A — 63° 39' -fl5° 32' 52" = 79° 11' 52", since A is oppo¬ 
site BC, the longer side. 

Sin. 79° 11' 52" : 161 = sin. 52° 42' : AB. 

Co. log. sin. 79° 11' 52" = 0.007764 
Log. 161 = 2.206826 

Log. sin. 52° 42" = 9.900626 

2.115216 = \o<r.AB = 130.38. 

Arts. 130.38 yards. 

2. To ascertain the distance between objects A and B , lines 
are measured to station C, 178 and 212 feet; the horizontal 
angle is found to be 61° 40'. What is the distance ? 

Ans. 202.01 feet. 

3. From two stations, A and B , distances are measured to 
a third station C, 97 and 86 yards; the angle between A and 
B at C is 31° 50'. Required distance from A to B. 

4. Wishing to know the distance between two points separ¬ 
ated by swamp and wood, I measure to a station distances 
from both, and find them 120 and 133 yards; the angle I find 
to be 57° 10'. What is the distance? 


Case 2. When the objects are not easily accessible. 


Measure a base line from each extremity of which both 
objects are visible; at each extremity measure the horizontal 
angle between each object and the other station. 


Then if C and D be the objects, A and B 
the stations, there are known AB , base line, 
the angles CAB , BAB , ABB and ABC. 

Also, CAB = CAB-BAB (1) 

and CBB = ABB-ABC, (2) 

and they are known. 

Also, ACB = 180 (CAB+ABC) 

and ABB = 1$Q 0 -(ABB+BAB ), 

and they are known. 


c 


D 



( 3 ) 





102 SURVEYING AND NAVIGATION. 

Therefore, in the triangle ABC\ (Chap. 2, Sec. 2, Prop. IY), 
we may have 

Sin .ACB : AB = sin .CAB : BC, (5) 

And in ABB , 

Sin.AZbZ? \ AB — sin .DAB : BD. (6) 

In the triangle CBD , L CBD is known (2), and sides BC 
and BD (5 and 6). 

By Trig., Chap. 2, Sec. 2, Prop. VII. 

BC+BD : BC-BD=tzr\J 2 (BDC+BCD ): tan .^{BDC-BCD) (7) 
i(BD C+ B CD) + \(BDC—B CD)=BD C (8) 

And Sin .BDC: BC = sin. : Z>C, 

the distance required. 

By using the triangles ABD and ABC. to find AD and 
AC, the line CD may be calculated from the triangle ACD , 
as well as from BCD . 


EXAMPLES. 


1. A man desiring to ascertain the distance between two 
objects from which he is separated by a river, measures a base 
line 200 feet long. At one extremity, (station A), he finds 
the horizontal angles to be 83° 47', and 42° 32'; at the other, 
(station B) he finds the angles 76° 52’, and 36° 20'. What is 
the distance required ? 

Solution. 


If C and D represent the objects, A and B the stations, 
AB = 200 feet, CAB = 83° 47', DAB = 42° 32', ABD = 
76° 52', and ABC — 36° 20'. 


CAD = 83° 47'—42° 32' = 41° 15' 

CBD = 76° 52' -36° 20' = 40° 32' 

ACB == 180°—(83° 47' + 36° 20') = 59° 53' 
ADB = 180°—(76° 52'+ 42° 32') = 60° 36' 


( 1 ) 

( 2 ) 

( 3 ) 

0 ) 


MENSURATION. 


163 


fn triangle ABC\ 

Sin. 59° 53' : 200 = sin. 83° 47' : BC (5) 
Co. log. sin. 59° 53' = 0.062981 


Log. 200 : 

Log. sin. 83° 47' : 

Log. BC : 

= 2.301030 
= 9.997439 

= 2.361450 = log. 229.85 = BC. 

In triangle ABD , 



Sin. 60° 36’ : 200 = sin. 42° 32' : BD (6) 
Co. log. sin. 60° 36' — 0.059875 


Log. 200 : 

Log. sin. 42° 32' : 

Log. BD = 

= 2.301030 

= 9.829959 

= 2.190864 = log. 155.19 = BD. 

In triangle CBD , 



2(180°—40° 32') = 69° 44' = i(BDC+BCD) 
385.04 : 74.66 = tan. 69°44' : tan .\{BDC-BCD) (7) 


Co. log. 385.04 

Log. 74.66 = 

Log. tan. 69° 44' = 

Log. tan. 1( BDC—BCD) - 
^(B DC-BCD) = 
i(BDC+BCD) = 
BDC = 

= 7.414494 
= 1.873088 
= 10.432680 

= 9.720262 = log. tan. 27° 42' 18" 
= 27° 42' IS" 

= 69° 44' 

= 97° 26' 18". 

Again in CBD , 



Sin. 97° 26' 18" : 229.85 = sin. 40° 32' : CD. 
Co. log. sin. 97° 26' 18" = 0.003670 


Log. 229.85 

Log. sin. 40° 32' 

Log. CD 

= 2.361450 

= 9.812840 

= 2.177960 = log. 150.65. 


CD — distance required = 150.65 feet. 







164 SURVEYING AND NAVIGATION. 


2. Desiring to ascertain the distance between two objects, 
C and D , which are inaccessible, I select two stations, A and 
j B, 282 feet apart. At station A I find the angle between C 
and B 87° 50', between D and B 52° 40'. At station B I find 
angle between D and A 89° 34', between C and A 46° 24'. 
What is the distance sought ? Ans. 280.5 feet, nearly. 

3. Given the following: AB — 43 yards, CAB — 89° 45', 
DAB = 39° 40', ABD = 77° 10', and ABC = 29° 26', to 
find the distance CD. 


If it is not possible to find convenient stations from which 
both objects are visible, the distance between the objects may 
be determined by a series of triangular calculations. Thus, 

Suppose C and D c ^ 

to be two objects, so 
situated that both are 
not visible from any 
station. Take sta¬ 
tion A in sight of 67, 
and B in sight of D , 

A and B being in 
sight of each other. 

Measure as before, base line AB and angles BAC and ABD. 

Near A take the station E, from which both A and C are 
visible; measure AE, and the angles EAC and AEC; ACE 
is found at once by subtracting from 180°. Then 

sin. A CE: AE = sin .AEC: A C. (1) 

Aow, in the triangle ABC\ AB and AC are know r n, and 
the angle BAC ; and ^ve have 



AB+AC: AB—AC = tan.i (ABC+ACB) : 

tan .j(A CB—ABC), 
i (ABC+ACB) ± ACB—ABC = ABC and ACB. 
sin .ABC: AC = sin .BAC: BC. 


( 2 ) 

( 3 ) 

( 4 ) 



/ 


MENSURATION. 165 

• O- 

To find BD assume a station F, near B , from which B and 
D are visible; measure BF , BFD and FBI). As in trian¬ 
gle AEC ', 

sin .BDF : BF = sin .BFD : BD. (5) 

ABD is known by measurement; ABC is known from (3) 
ABD—ABC — CBD. From (4) and (5), we have BC and 
BD. We, therefore, have, as in the first part of the case, 
two sides and angle included to find the third side, which is 
the required distance. 


PROBLEM IV. 

To determine the vertical height of an object situated on an 
inclined plane. 

Method .—Measure the angle at the foot of the object 
between it and the plane; then meas¬ 
uring a convenient distance to a 
station, take the angle between 
the plane and the top of the ob¬ 
ject. 

Let AB be the object on an 
inclined plane. A C is known by 
measurement, as are also the angles 
BAC and ACB. Also, ABC= 

180°—(BAC+ACB). Hence, 

sin. ABC: AC — sin .ACB : AB = height required. 

EXAMPLES. 

1. To ascertain the height of a vertical object situated on 
an inclined plane, the angle at the base is measured and found 
to be 102° 42'. Proceeding 100 feet, the angle between the 
plane and the top of the object is found to be 41° 10'. What 
is the height of the object ? 








166 


SURVEYING AND NAVIGATION. 


Solution. 

In the triangle ABC, AC = 100, BAC = 102° 42 f , 
ACB = 41° 10', and ABC = 180°-(102° 42' + 41° 10') = 
36° 8'. 

sin. 36° S' : 100 = sin. 41° 10' : AB. 

Co. log. sin. 36° 8' = 0.229394 
Log. 100 = 2.000000 

Log. sin. 41° 10' = 9.818392 

Log. AB = 2.047786 = log. 111.63. 

AB = height = 111.6s feet. 

% 

2. To determine tlie height of a tower on an inclined sur¬ 
face, there are measured the angle at the base, 107° 35', and 
the angle at an assumed station, 41° 54' ; also, the distance 
from base to station, 80 feet. What is the height ? 

Ans. 105.21 feet. 

3. To determine the height of a tree upon a hillside, I 

measure at the foot of the tree the angle with the plane of the 
hill, 105° 50', and at a station distant 40 feet, 51° 12b Re- 
quired the height of the tree. Ans. 79.89 feet. 


2 d Method .—Measure as before from the foot of the object 
to a station, and then take the angle between the plane and 
top of object. Then measure in a 
line directly from the object to a 
second station, and then take an an¬ 
gle as before. 

In the triangle BCD , C and D 
being the two stations, and AB the 
object, there are known DC, BDC, 
and BCD = 180° — BCA ; also, 

CBD = ISO °-(BCD+BDC). 



Then, 


sin. CBD : DC = sin.BDC: BC 


(1) 







MENSURATION. 


167 


In triangle ABC ’, BC is known (1); also, AC and ACB 
by measurement. 

BC+AC: BC—AC = tan .&CAB + ABG) : 

t&w.^CAB—ABC) (2) „ 

Whence, ABC and BAC are to be found, 

An. ABC : AC — sin. ACB : AB. (3) 

Note. —To obtain the angles at C and D, the angles of elevation should 
be taken, and from them the angle of inclination of the plane subtracted. 


EXAMPLES. 

1. To determine the height of a tower situated on a hill¬ 
side inclined 27° 48' 11", I measure 50 feet from the base of 
the tower, and take the angle of elevation, 67° 58' 11"; then 
measuring in the same direction 100 feet, I find the angle of 
elevation 48° 0' 11". What is the height of the tower? 

Solution . 

AC = 50, CD = 100, ACB = 67° 58' 11"—27° 48' 11" = 
40° 10', CDB = 48° 0' 11"—27° 48' 11" = 20° 12'; also, 
BCD = 180°—40° 10' = 139° 50', and CBD = 40° 10'— 
20° 12' = 19° 58'. 

In the triangle BCD , 

sin. 19° 58' : 100 = sin. 20° 12' : BC. (1) 

Co. log. sin. 19° 58' = 0.466643 
Log. 100 - 2.000000 

Log. sin. 20° 12' = 9.538194 

Log. BC = 2.004837 = log. 101.12. 



168 


SURVEYING AND NAVIGATION. 


In triangle ABC\ 

&CAB+ABC) = |(180 o —40° 10') = 69° 55'. 

151.12 : 51.12 = tan. 69° 55' : tax\.^{CAB—ABO). (2) 

Co. log. 151.12 = 7.820678 
Log. 51.12 = 1.708591 

Log. tan. 69° 55' = 10.436972 

Log. tan .\{CAB-ABC) = 9.966241 = tan. 42° 46' 31". 

69° 55'—42° 46' 31" = 27° 8' 29" = ABC, lesser angle, 
opposite A C. 

sin. 27° 8' 29” : 50 = sin. 40° 10' : AB (3) 

Co. log. sin. 27° 8' 29” = 0.340856 
Log. 50 = 1.698970 

Log. sin. 40° 10' = 9.809569 

Log. AB = 1.849395 = log. 70.696. 

A ns. = 70.696 feet. 

2. From an object on a plane inclined 15° 30' 57”, having 
measured 66 feet, the angle of elevation was 66° 10' 57”; 100 
feet farther on, the angle of elevation was 46° 52' 57”. What 
was the height of the object ? 

Ans. 126.41 feet. 

3. Given, distance from base of object to first station 70 
feet, and angle at first station 46° 40'; distance from first to 
second station 45 feet, and angle 38° 18'. Bequired height 
of object. 


PROBLEM V. 

To determine the horizontal distance from any point to an 
inaccessible object. 

Measure from the given point abase line to some convenient 




MENSURATION. 


109 


station, and at each point measure 
the horizontal angle between the ob¬ 
ject and the other station. 

Let A be the object, and B the 
station, from which the distance is to 
be found. Then BC being measured, 
and the angles i? and <7, we have CAB 
= 180°— (B+C); and by Plane Trig., Chap. 2, Sec. 2, 
Prop. IY. 

sin.^1 ; BC = sin. C: AB , the required distance. 

EXAMPLES. 

1. Desiring to ascertain the distance between two houses on 
opposite sides of a river, from one I measure to a station (7, 
100 yards ; and measure also the angle at first station, 72° 4P; 
at the second, C f 83° 35h 


A 



Solution, 

* 

BC = 100, L^ = 72°41', C — 83° 35' A = 180°- 
156° 16' = 23° 44'. 

Sin. 23° 44' : 100 = sin. 83° 35' : AB. 

Co. log. sin. 23° 44' = 0.395255 

Log. 100 = 2.000000 

Log. sin. 83° 35' = 9.997271 

Log. AB = 2.392526 = log. 246.9 

Am. 246.9 yards. 

2. To find tlie distance between two objects separated by a 
stream, a base line is measured 210 feet to a station. The 
horizontal angles are then measured ; at the object 81° 40', 
at the station 70° 10'. What is the distance? 

Am. 418.48 feet. 



8 




/ 


170 PURVEYING / ND NAVIGATION. 

3. Given a base line BC — 160 feet, and angles, B = 
89 ° 3', C — 57° 56', to determine distances from B and C to 
an object A. 


PROBLEM VI. 

Given the distances bet ween three points , and also the angles 
between these points from a distant object , to determine the dis 
tancefrom the distant object to the three points respectively. 

This problem may be best understood by a careful examina¬ 
tion of a practical example. 

Coming from sea, at the point D I observed two headlands, 
A and B , and inland at C a steeple which appeared between 
the headlands. I found, from a map, that the headlands were 
5.35 miles from each other; that the distance from A to the 
steeple was 2.8 miles, and from B to the steeple 3.47 miles ; and 
I found, with a sextant, that the angle ADC was 12° 15 f , and 
the angle BBC , 15° 30h Required my distance from each of 
the headlands, and from the steeple. 

Construction . 

The angle between the two headlands is 
the sum of 15° 3b' and 12° 15 f , or 27° 45 1 
Take double this sum, 55° 30h Conceive 
AB to be the chord of a circle, and the 
arc on one side of it to be 55° 30'; and, of 
course, the other will be 304° 30'. The 
point D will be somewhere in the circum¬ 
ference of this circle. Consider that point as determined, and 
draw CD. 

In the triangle ABC we have all the sides, and of course 
we can find all the angles; and if the angle ACB is less than 
180°—27° 45' = 152° 15 f , then the circle cuts the line CD in 
a point B] and C is without the circle. 


c 




MENSURATION. 


171 


Draw AE\ EE, AD, and ED. AEED is a quadrilateral 
in a circle, and \_AEE-\- L ADE = 180°. 

The L ADE — the L ABE, because both are measured by 
one half the arc AE. Also, [_ EDE = L EAE, for a simi¬ 
lar reason. 

Now, in the triangle AEE, its side AE, and all its angles, 
i e known ; and from thence AE can be computed. Then, 
having the two sides, AC and AE, of the triangle AEC, and 
the included angle CAE, we can find the angle AEC, and of 
course its supplement, AED. Then, in the triangle AED, 
we have the side AE, and the two angles AED and ADE, 
from which we can find AD. 


Solution. 


The computation, at length, is as follows: 


To find AE. 


Angle EAE — 15° 30 f 

Sin. A AT?, 152 t ' 15'. 

, 9.668027 

Angle EBA = 12° 15' 

: AB, 5.35, 

.728354 

27° 45' : 

:: sin. AEE, 12° 15 ? , 

9.326700 

180° 


10.055051 

Angle AED = 152° 15' 

: AE, 2.438, 

.387027 

To find the 

angle EA C. 


BC, 3.47 



AE, 5.35 

log. .728354 


AC, 2.80 

log. .447158 


2)11.62 

1.175512 


S, 5.81 

log. .764176 


S -BC, 2.34 

log. .369216 



20 _ 

21.133392 

2)1 9.95788 0 fForward. 















I 


172 SURVEYING AND NAVIGATION. 






17° 41' 5S" 
2 

Angle BA C — 35~°~23'T6" 
Angle EAB = 15° 30' 

Angle EAC = 19° 53' 56" 

180° 

2 )160° 6' 4" 
80° 3' 2" 


2)1 9.957880 
cos. 9.978940 


AEC + ACE 
2 


To find the angles AEC and A CE. 


AC+AE 
: AC—AE 

AEC+ACE 


:: tan. 


2 


: tan. 


angle AEC, 


AEC-ACE 


5.238 

.362 

80° 3' 2" 


21° 30' 12" 


101 a 33' 14" , sum, 

angle A CE or A CD, 58° 32' 50", diff. 
angle CD A, 12° 15' 


719165 

1.558709 

10.755928 

10.314637 

9.595472 


70° 47'~50", supplement 109° 12' 10'', angle CAD 

35° 23' 56" , angle CAB 

73° 48' 14", angle BAD 


To find AD. 


Sin .ADC, 12° 15', 9.326700 

: AC ; 2.8, .447158 

:: sin .A CD, 58° 32' 50", 9.930985 

10.378143 

: AD , 11.26 miles. 1.051443 

.CD and BD may also readily be found. 

f AD = 11.26, 
Ans. •] BD = 11.03, 
( CD = 12.46. 

























MENSURATION. 


173 


The preceding problems include those most frequently met; 
others may arise, requiring different construction ; but an 
acquaintance with the principles involved in the problems 
here given, and a knowledge of geometrical constructions and 
relations will soon give a key to any question arising. The 
miscellaneous examples which follow require mainly only the 
rules and constructions given in this section ; whatever is re¬ 
quired further the student will see and work out for himself 
with little difficulty. 


PRACTICAL PROBLEMS. 


1. Required the height of a wall whose angle of elevation, 
at the distance of 463 feet, is observed to be 16° 21'. 

A?is. 135.8 feet. 

2. The angle of elevation of a hill is, near its bottom, 31° 
18 f , and 214 yards further off, 26° IS'. Required the perpen¬ 
dicular height of the hill, and the distance of the perpendic¬ 
ular from the first station. 

/ The height of the hill is 565.2 yards, and the dis- 

Ans. •< tance of the perpendicular from the first station is 
( 929.6 yards. 

3. The wall of a tow T er which is 149.5 feet in height, makes, 
with a line drawn from the top of it to a distant object on the 
horizontal plane, an angle of 57° 21h What is the distance 
of the object from the bottom of the tower ? 

Ans. 233.3 feet. 


4. From the top of a tower which is 138 feet in height, I 
took the angle of depression of two objects standing in a direct 
line from the bottom of the tower, and upon the same horizon¬ 
tal plane with it. The depression of the nearer object was 
found to be 48° l(f, and that of the further, 18° 52 f . What 
was the distance of each from the bottom of the tower ? 

A ( Distance of the nearer, 123.5 feet: 

Ans. < 

{ and of the further, 403.8 feet. 


174 SUKVEYING AND NAVIGATION. 


5. Being on the side of a river, and wishing to know the 
distance of a house on the opposite side, I measured 312 yards 
in a right line by the side of the river, and then found that the 
two angles, one at each end of this line, subtended by the 
other end and the house, were 31° 15' and 86° 27'. What 
Was the distance between each end of the line and the house ? 

Ans. 351.7, and 182.8 yards. 

6. Having measured a base of 2G0 yards in a straight line 
on one bank of a river, I found that the two angles, one at 
each end of the line, subtended by the other end and a tree on 
the opposite bank, were 40° and 80°. What Avas the width 
Vf the river ? 

Ans. 190.1 yards. 

7. From an eminence of 268 feet in perpendicular height, 
the angle of depression of the top of a steeple, which stood on 
the same horizontal plane, was found to be 40° 3', and of the 
bottom, 56° 18'. What Avas the height of the steeple \ 

Ans. 117.76 feet. 

8. Wanting to know the distance between two objects which 
were separated by a morass, I measured the distance from each 
to a point from whence both could be seen; the distances were 
1840 and 1428 yards, and the angle which, at that point, the 
objects subtended, Avas 36° IS' 24". Required their distance. 

Aiis. 1090.85 yards. 

9. It is required to find the distance from a toAver, 80 feet i:i 
height, to an object Avliose angle of depression from the top 
Df the tower is 22° 41 f . 

4 

Ans. 191.4 +feet. 

10. The angle of elevation of a hill from a station near its 
foot is 29° 28'; from a station distant 100 yards, the angle of 
elevation is 20° 10' 30". Required perpendicular height of 
hill, allowing 5 feet for height of instrument. 

Ans. 106.74 yards nearly. 


MENSURATION. 


175 


11. From two stations, 300 feet apart, the horizontal angles 
made with a distant church are taken 88° 19' and 89° 40'. 
Required distances from church to stations. 

Ans. 8521.4 and 8525 feet nearly. 

12. Wishing to know the distance between two inaccessible 
objects, C and D, I take two stations, A and B , 245 i'eet apart, 
and measure the angles as follows : BAC — 80° 13', BAD = 
41° IF, ABB = 80° 37', and ABC = 35° 9'. Required 
distance from C to D. 

Ans. 204.39 feet. 

13. Required the height of a wall whose angle of elevation, 

100 feet from its base, is 18° 26'. Ans. 33.33 feet. 

14. Wishing to know the height of an inaccessible object, I 
measure a base line AB, 190 feet, and take the horizontal 
angles; at A, 56° 40'; at B , 63° 11'. I measure also the angle 
of elevation at A — 28° 21'. Required the height of object. 

A ns. 105.45 feet. 

15. It being desired to ascertain distance between two 

houses, separated by swampy ground, a convenient station is 
selected and distances measured 472 and 560 feet. The angle 
between the houses at the station is found to be 63° 14'. 
What is the distance? Ans. 546.25 feet. 

16. Wanting to know the breadth of a river, I measured a 
base of 500 yards in a straight line on one bank; and at each 
end of this line I found the angles subtended by the other end, 
and a tree on the opposite bank of the river, to be 53° and 79° 
12'. What was the perpendicular breadth of the river? 

Ans. 529.48 yards. 

17. What is the perpendicular height of a hill, its angle of 
elevation, taken at the bottom of it, being 46°, and 200 yards 
further off, on a level with the bottom, 31° ? 

Ans. 286.28 yards. 


176 


SURVEYING AND NAVIGATION. 

18. Wanting to know the height of an inaccessible tower, 
at the least accessible distance from it, on the same horizontal 
plane, I found its angle of elevation to be 58 a ; then going 300 
feet directly from it, I found the angle there to be only 32°; 
required the height of the tower, and my distance from it at 
the first station. 

j Height, 307.54 feet. 

1 Distance, 192.18 u 


19. Two ships of war, intending to cannonade a fort, are, 
by the shallowness of the water, kept so far from it, that they 
suspect their guns cannot reach it with effect. In order, 
therefore, to measure the distance, they separate from each 
other a quarter of a mile, or 440 yards, and then each ship 
observes and measures the angle which the other ship and fort 
subtends ; these angles are 83° 45', and 85° 15'. What, then, 
is the distance between each ship and the fort ? 

Am. \ 2292 - 2S " ards - 
I 2298.05 “ 

20. From two ships, A and B , which are anchored in a 
bay, two objects, C and A>, on the shore, can be seen. These 
objects are known to be 500 yards apart. At the ship A , the 
angle subtended by the objects was measured, and found to be 
41° 25'; and that by the object D and the other ship was 
found to be 52° 12 f . At the other ship, the angle subtended 
by the objects on shore was found to be 48° 10 r ; and that by 
the object C, and the ship A y to be 47° 40*. Required the 
distance between the ships, and the distance from each ship to 
the objects on shore. 

Distance between ships, 395.7 yards. 
From ship A to object D , 743.5 
From ship A to object C y 467.7 
From ship B to object A), 590.5 


Ans . 


a 


u 


a 


To solve this problem, suppose the distance between the 


\ 




MENSURATION. 


177 


ships to be 100 yards, and determine the several distances, in¬ 
cluding the distance between the objects, C and D , under this 
supposition ; then multiply the values thus found for the 
required distances by the quotient obtained by dividing the 
given value of CD by the computed value. 

Full solutions of the Examples and Problems of this entire work may be found 
i i the Key to Robinson’s Geometries and Surveying. 


SECTION II. 

MENSURATION OF SURFACES. 

The Area of a figure is the surface included between the 
lines which bound it. 

Strictly, for the measurement of areas a superficial unit 
should be applied ; but as all ratios are abstract, for practical 
convenience the ratio of lines is used—the quality of superfi¬ 
cies being attached to the result. 

A superficial unit is generally the square formed upon a 
linear unit of the same name. Thus, the square inch and 
square yard correspond to the linear inch and linear yard, 
which are the sides of the superficial units. 

Roods and acres, units used in measuring land, have no cor- 
responding linear units. 

The following are the tables of superficial units. 


SQUARE MEASURE. 


144 square inches (sq. in.) 

9 square feet 
30[ square yards 
40 square rods 
4 roods 
640 acres 


make 

1 

square foot, . 


make 

1 

square yard, 

.... sq. yd. 

make 

1 

square rod, . 

.... sq. rd. 

make 

1 

rood,. 

.R. 

make 

1 

acre,. 

.A. 

make 

1 

square mile, 

. . . . sq. mi. 








178 


SURVEYING AND NAVIGATION. 


surveyors’ square measure. 


625 

square links (sq. 1.) 

make 

1 pole,. 

. . . P. 

16 

poles 

make 

1 square chain, . . . 

sq. ch. 

10 

square chains 

make 

1 acre,. 

... A. 

640 

acres 

make 

1 square mile,.... 

sq. mi. 

36 

square miles (6 miles square) 

make 

1 township,. 

. . Tp. 

The following rules for the measurement of surfaces being 


mainly founded upon Geometry, the student is referred for 
tlieir demonstration to Robinson’s New Geometry, the num¬ 
ber of the Book and Proposition being given with each rule. 


PROBLEM I. 

To find the area of a parallelogram. 

Rule 1 . — Multiply the base by the altitude. (Geom. Def. 54 
—B. I., Th. 27.) 


EXAMPLE. 


Required the area of a parallelogram whose base is 23 and 
altitude 11 feet. 

23x11 = 253. .*. Area required = 253 square feet. 

Note. —To illustrate the transfer from linear to superficial units, let A = 
unit of measure, a square, having each side unity ; and C, equal the rectan¬ 
gle equal in area to the parallelogram to be measured. Then from Def. 54, 
Geom., we may conclude, letting a and b represent altitude and base of C. 


A : C =■ \ x \ : axb, or 

(1) 

1 : C = 1 x 1 : a x b. .*. 

(2) 

C axb 

(3) 

1 ~ 1 x 1 


If for a and b , we place 11 and 23 a3 in above example, we have 

C 11x23 

T = TxT 


= 253. 


G) 







MENSURATION. 


179 


That is, the unit rectangle is contained as many times in the given rec¬ 
tangle or parallelogram, as 1 x 1, is contained times in 11 x 23; that is, 253 
times. Hence 253 = 23 x 11, expresses the ratio between the unit square 
foot and the surface measured; or the given surface contains one square fo >t 
253 times, or 253 square feet. Here then we have two ratios—abstract 
numbers and equal, obtained, one by applying superficial unit to surface, the 
other by applying linear units to lines. We use the more convenient (i 
these two ways of obtaining the ratios, and since the two are equal, v q 
name and consider the result of one process as though it were the result of 
the other. 

Rule 2.— When two sides and the angle included are 
known .* Multiply the product of the sides by the sine of the 
included angle . 


Demonstration . 

Let ABCD be the 
parallelogram, AB its 
base, and DE its alti¬ 
tude. 

(Rule 1.) 

Area — AB x DE. (1) 

(Plane Trig., Chap. 

2, Sec. 2, Prop. Ill, 

B : AD = sin. A : DE, (2) 

I) j_- UX>xsin.U (3) 

B 

Substituting for DE from (3) into (1), 

. AB x AD x sin.A 

Area = - n - 

B 

When B= 1, Area = AB x AD x sin .A. 

Tf logarithms are used in the calculation, B must be con¬ 
sidered, its logarithm being 10. 








180 


SURVEYING AND NAVIGATION. 


EXAMPLES. 


1. The sides of a parallelogram being 42 and 18 feet, and 
the angle included 41° 11', required the area. 

Calculation . 


Area 


42 x 18 x sin.41° ll f 
~~R 


1.623249 

1.255273 

9.818536 


Log. 42 
Log. 18 

Log. sin. 41° 11' 

Sum less 10 (= log. A) = 2.697058 = log. 497.8=area. 


2. Required the area of a parallelogram whose base is 11 

ft. 3 in., and altitude 10 ft. 6 in. Ans. 118| sq. ft. 

3. Required the area of a parallelogram whose sides are 31 
•and 11 feet, and included angle 31° 18'. 


PROBLEM I 1. 

To find the area of a triangle . 

Rule 1 . — Take one half of the product of the base and 
altitude. (Geom., B. I, Th. 33.) 

Rule 2.—When two sides and angle included are known ; 
Take one half of the product of the two sides by the sine of 

the included angle , divided by radius. (See Rule 2, Prob. 1.) 

§ 

EXAMPLES. 


1. What is the area of a triangle whose base is 36 and alth 


tude 11 ? 


Ans. 


36*11 


198. 


2 





MENSURATION. 


181 


2. What is the area of a triangle two of whose sidefe are 

45 and 31 inches, and the angle included 47° 39' ? 

a 45*31 sin* 47° 39' kikaq • i 
Area =-— - = 515.48 so. inches, 

2*i2 H ’ 


A ns. 3 ft. 83.48 inches. 


3. What is the area of a triangle, having sides 12 and 9 
feet, and angle included 56° 20' ? 

Ans. 44.94 feet. 

4. What is the area of a triangle whose sides are 42 and 
60 feet, and the included angle 61° 12' \ 


Rule 3. —When the three sides A a triangle are known: 
From one half the sum of the sides , subtract each side separ¬ 
ately * multiply the continued product of these remainders by 
the half sum / the square root of the product will be the area 
required . 


Demonstration . 


Let A represent the area of a triangle 
ABC , whose sides &, b , and c are known ; 
a being considered as the base. Draw 
AD perpendicular to CB ; it will be the 
altitude. 

By (Geom., B. I., Th. 41.) 

b 2 = a 2 Ac 2 —2 axBD 
a 2 Ac 2 — b 2 


BD =. 


2 a 



By (Geom., B. I., Th. 39), AD 2 = c 2 - BD 2 , 
Substituting from (2) into (3), the value ot BD, 

(a 2 Ac 2 — b 2 ) 2 
4m 2 


AD 2 = c 2 -- 


A 7^ _ Via 2 c 2 —(a 2 Ac 2 —b 2 Y 

An - -2S 


( 1 ) 

( 2 ) 

( 3 ) 

( 4 ) 

( 5 ) 


/ 










182 


SURVEYING AND NAVIGATION. 


By Rule 1 , A = ( 6 ). 

Substituting in ( 6 ) the value of from (5), 


V4a 2 c 2 — (a 2 +c 2 — b 2 ) 2 ./4a 2 c 2 — (a 2 4-c 2 — b 2 ) 2 

=-4-=V-'is:- (7) 

Factoring, 



Since tlie difference of the squares of the two quantities 

equals the product of their sum and difference. Let • 

2 

,i b-c—a 7 a-\~c—b 7 &~\~b—c 

then s—a — — -, s—b — —!-, and s—c = - T 

2 ’ 2 ’ 2 ' 
Substituting these values in ( 8 ), 


A = Vs(s—a)(s—b)(s—c). 
Whence the rule. 




EXAMPLES. 

1. Required the area of a triangle whose sides are 342, 384 
and 436 feet. 


Solution. 


a = 436 

s—a = 145 ; log. = 2.161368 

h = 384 

s—b = 197 ; “ = 2.294466 

c = 342 

s—c = 239 ; “ = 2.378398 

2 ) 1162 

= 581; “ = 2.764176 

^ = 581 

2 ) 9.598408 

4L 


Log. A = 4.799204 


4.799204 = log. 62980.14 + 

Area 62980.14 sq. ft. 
























MENSURATION. 


1S3 


2 . How many square yards in a triangle whose sides are 
78, 82,and 100 feet? 

Ans. 34.6 + sq. yards. 

3. Required the area of a triangle whose -sides are 31, 40, 
and 55 rods. 

Ans . 3.8 acres. 

PROBLEM III. 

To find the area of a trapezoid. 

Rule.— Multiply one half the sum of the parallel sides by 
the altitude. (Geom. B. I., Th. 34.) 


EXAMPLES. 

1 . What is the area of a trapezoid whose parallel sides are 
23 and 11 feet, and whose altitude is 9 feet ? 

23 -f 11 


2 


x9 = 153. 


Ans. 153 sq. ft. 


2. Required the area of a trapezoid whose parallel sides are 
178 and 146 feet, and whose altitude is 69 feet. 

Ans. 41.05 sq. rods. 


3. How many acres are there in a trapezoid whose bases 
measure 38 and 26 rods, and altitude 10 rods ? 


PROBLEM IV. 

To find the area of a trapezium,. 

Rule 1 . —If the sides and two opposite angles are known :— 

Multiply the sine of each angle [of the two opposite ) by one 
half the product of the sides which include it. The sum of the 
two products so obtained will be the area required. 



184 


SURVEYING AND NAVIGATION. 


Demonstration. 

Let A BCD be the trapezium whose 
sides are known, and also two angles, 
as A and C. 

By Prob. II, Rule 2, 

Area of ABD = sin.A x - 
Area of BCD — sin .C X- 

Hence, 

ABCD = ABD+BCD = sin .Ax 

sin.< 

Rule 2. —If only the sides are known, a diagonal must be 
measured ; two triangles will thus be formed, whose areas may 
be found by Rule 3, Problem II. The sum of these areas will 
be the area of the trapezium. 

Rule 3.— Without determining the sides, a diagonal may 
be measured, and also perpendiculars from the opposite angles 
upon that diagonal. In the two triangles formed by the diag¬ 
onal, there will then be known the base and altitude, and 
Rule 1, Problem II, may be used. 



ABxAD 

2 


+ 


7x BCx fD (3) 


EXAMPLES. 

1 . Required the area of a trapezium whose sides, AB and 
AD, are 32 and 17 feet, and the angle A 71° 10 f ; sides CB 
and CD, 30 and 13 feet, and the angle C 108° 53'. 








MENSURATION. 


1S5 


Solution . 


Drawing diagonal BD by Rule 1 st, we have, 

32 x IT x sin. Tl° 10 ' 


Area ABD ~ 


Log. 32 

IT 


a 


Log. sin. Tl° 10 ' 


Log. 2 B 
Log. area 


2 R 

= 1.505150 
= 1.230449 
= 9.9T6103 


12 .T 11 T 02 

10.301030 


= 2.4106T2 == log. 25T.44. 


(i) 


Area CBD = 30 x 13 x sin. 108° 53 1 ^ 

2 R v 

Log. 30 = 1.477121 

“ 13 ' = 1.113943 

Log. sin. 108° 53' = 9.975974 

12.567038 
Log. 2 R = 10.301030 

Log. area = 2.266008 = log. 184.5. 

Area ABD = 25T.44 
Area CBD = 184.50 

441.94 = area of trapezium. 


2. Required the area of a trapezium, whose sides are 9.5, 
11 , 12 and 14.8 rods, and whose diagonal from first to third 
station is 18 rods. 

Ans. 132.25 sq. rods. 

3. Required the area of a trapezium, whose diagonal meas¬ 
ures 1T.5 rods, and the perpendiculars from the angles upon 
that diagonal 8.4 and 4 rods. 


Ans. 108^ sq. rods. 









186 


SURVEYING AND NAVIGATION. 


Another method still may he 
employed, as follows: let ABCD 
he the trapezium. At any angle 
as A, draw AM perpendicular to 
AB. From angles C and D draw 
lines CE and DF perpendicular 
to AM, and thus parallel to AB. 

Then ABCE and ECJDF will he 
trapezoids, and ABE a right- 
angled triangle. Having known 
AB, EC and FD, and also AF, 

AE and EE, the areas of these figures may he found hy 
Prob. II., Pule 1, and Prob. III. 

Area ABCD — ABCE+ECDF-ADF. 

Whence is determined the area required. 

This process is the one employed in rectangular surveying, 
and may he applied to all polygons. 



! 


PROBLEM V. 

To find the area of an irregular polygon. 

Pule. —Divide the polygon into triangles by diagonals, and 
draw perpendiculars from the vertical angles of these triangles 
upon the diagonals. Having measured the diagonals and 
perpendiculars, determine the areas of the triangles. The sum 
of these areas will be the area of the polygon. 

EXAMPLE. 

To determine the area of the polygon 
EABCD, I measure the diagonals EB 
and EC, 60 and 68 feet. The perpendic¬ 
ulars I find to be as follows: from A, 10 
feet; from B, 30 feet; from D, 25 feet. 

What is the required area? 








MENSURATION. 


187 


Solution . 


Prob.II., Rule 1, 60x10 = 300 = EAB. 

’ 2 

68x30 = 1020 = EEC. 

2 

68 x - = 850 = EEC. 

2 _ 

Sum = 2170 = area EABCD. 


2. What is the area of an irregular polygon whose diago¬ 
nals are 32 and 56 feet, and perpendiculars as follows: upon 
the first diagonal, 7; upon the second, 11 and 13 feet. 

Ans. 87^ sq. yards. 


PROBLEM Y I. 

To find the area of a regular polygon. 

Rule. —Multiply one half the perimeter hy the perpendic¬ 
ular drawn f rom the center upon one of the sides. 

Demonstration. 

Let ABCDEFbe a regular poly¬ 
gon, whose center is M. From M 
draw MA, MB, &c., and let fall 
MM perpendicular to AB. MM 
will be the altitude of the triangle 
ABM, and by Prob. IT., Rule 1, 

Area ABM — \AB x MM. 

M, being the center of the polygon (Geom., B. IV, Th. 36, 
Cor. 2 ), it is equidistant from the sides: that is, MN is equal to 
any perpendicular from M upon the sides, and may represent 
the common altitude of the triangles ABM, BMC , &c. Hence, 


E T> 















188 


SURVEYING AND NAVIGATION. 


the areas of the triangles having for bases BC\ CD, &c ., will 
equal each one half its base into MN ; and the area of the poly¬ 
gon, which equals the sum of the triangles, will equal 

MT v (AB + BC+ CD+DE+ EE A FA) 

2 

Whence the rule. 

EXAMPLES. 

1 . What is the area of a regular hexagon whose side is 
8 feet, and the perpendicular 6.92 feet. 

_ 4.Q 

Perimeter = 8 x 6 = 48, ~ x 6.92 = 166.08 square feet. 

If the perpendicular is not known, it may be determined 
from the following proportion, (Chap. 2 , Sec. 2 d. Prop. Ill), 

MN : AN — It : tan .AMN. (See preceding figure.) 

For (by Geom., B. IY., Th. 30, Cor. 3), 

The angle AMB = __ 36Q °_ 

No. ot sides ot polygon, 

And AMN is then known, being one half of AMB , since 
the triangle AMB is isosecles; and for the same reason AN 
is known, being one half of AB. Hence, AN and AMN 
being known, from the above proportion MN may be found. 

2 . What is the area of a regular polygon of eight sides, each 
side being 6 feet ? 


Solution. 

AMN = \AMB = i = 22° 30', and AN =^AB = Z. 

MN : 3 = R : tan. 22° 30'. 

Log. 7?+log. 3 = 10.477121 

Less log. tan. 22° 30' = 9.617224 

Log. MN = 0.859897 = log. 7.243. 

Area = |(6 x8)x 7.243 - 173.8 sq. ft., Ana. 






MENSURATION. 


189 


3. Required the area of a regular nonagon W jse side is 
12 feet. 


Ans. 890.18 sq. ft. 

4. Required the area of a regular pentagon whose side is 
3 feet. 

Ans. 15.48 sq. ft. 

Rule 2.— Multiply the area of a regular polygon of ti,e 
same number of sides , and each of whose sides is unity , by the 
square of one side of the required polygon. 

For if P represent the polygon whose area is required, and 
a one of its sides ; also, p the polygon whose side is unity, we 
shall have (Geom., B. II., Th. 22), 

p : P — l 2 : <z 2 , or 
P — . Whence the rule. 


In the use of the above rule, the following table, giving the 
area of the polygons when the sides are unity, with their log¬ 
arithms, will be found serviceable. 


TABLE. 


NAMES. 

SIDES. 

AREAS. 

LOGARITHMS. 

Triangle. 

3 

0.4330127 

1.6365007 

Square. 

4 

1.0000000 

0.0000000 

Pentagon. 

5 

1.7204774 

0.2356490 

Hexagon. 

6 

2.5980762 

0.4146519 

Heptagon. 

7 

3.6339124 

0.5603744 

Octagon. 

8 

4.8284271 

0.6838057 

Nonagon. 

9 

6.1818242 

0.7911166 

Decagon. 

10 

7.6942088 

0.8861640 ' 

Undecagon. 

11 

9.3656399 

0.9715375 

• Dodecagon. 

12 

11.1961524 

1.0490687 





























190 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

1. What is the area of a regular pentagon whose side is 

4 feet ? 

From the table, pentagon whose side is 1 = 1.7204774 

Multiply by 4 2 = 16 

Area recpiired = 27.5276384 

2. Required the area of a regular octagon whose side is 

5 feet. Ans. 120.71 sq. ft. nearly. 

3. Required the area of a regular heptagon whose side is 
7 feet. 

PROBLEM VII. 

To determine the circumference of a circle from the radius 
or diameter. 

Rule.— Multiply the diameter by 3.14159. (Geom., B. Y., 

Th. 6.) 

As 7 t is always used to express the above 3.14159, the rule 
may be given analytically, 

C = 27 tD. 

% 

EXAMPLES. 

1. What is the circumference of a circle whose radius is 
5 feet? 

* 2. What is the circumference of a circle whose diameter is 
18 feet ? 


PROBLEM VIII. 

To determine the diameter of a circle from the circumference. 

Rule. — Divide the circumference by 3.14159/ or multiply 
by .31831. 

From Prob. YU, C = 27rA 2 R = —. 

7T 





MENSURATION. 191 

EXAMPLES. 

1. Required tlie diameter of a circle whose circumference 

is 39.8 feet. Ans. 12.67 feet, nearly. 

2. What is the radius of a circle whose circumference is 
21.37 inches? 

3. What is the diameter of a circle whose circumference is 
137.81 feet. 


PROBLEM IX. 

To determine the area of a circle. 

Rule 1 . —Multiply the circumference by one half the radius. 
(Geom. B. Y, Th. 1.) 

Rule 2.— Multiply the square of the radius by 3.14159. 
For by Prob. VII., C = 27rT?, 

By Rule 1, Area — Cx—. 

2i 

Area = 2nd? x ~ = ttR 2 

Jj 

which is the analytical expression of the rules. 


EXAMPLES 

1. What is the area of a circle whose radius is 9 feet ? 

Ans. 9 x 9 x 3.14159 = 254.47 sq. ft., nearly. 

2. What is the area of a circle whose diameter is 12 rods ? 

3. What is the area of a circle whose radius is 11 feet? 


192 


SURVEYING AND NAVIGATION. 


PROBLEM X. 


To determine the length of a circular arc of any number 
of degrees . 

Rule .—Multiply the circumference of the circle by tic 
ratio between the number of degrees in the arc , and 360 . 

For (Geoiru B. !•? Bef. 53, and Th. 2, Cor. 2), 

360° : circumference = number of degrees in arc : arc 

No. of degrees in arc 
Arc = circumference x -rr^o- 


A. If the cliord of the arc be 
given, and its height, the diame¬ 
ter may be readily found. 

For (Geom. B. III., Th. 17, Cor.; 
AB being the chord, and DE the 
height of the arc, we have 
DE' DE = AD* 

AD* 


and 


DE -_ 

DE 

EE = DF+DE 



B. If the diameter of the circle is known, and either the 
chord or height of the arc, the number of degrees in the arc 
may be determined. 

By (Trig., Chap. 2, Sec. 2, Prop. Ill), 

R : AC — o,o§.ACD : CD (1) 

R : AC = sm.ACD : AD (i) 

When AC = radius of the circle, CD = radius less the 
height of the arc, and AD = one half the chord; ACD being 
one half the angle subtended by the chord. From (1) having 
radius of circle and height of arc, ACD may be found; ana 
from (2) having radius AC and chord AB , the same angle 
becomes known. Hence, w T e find the half arc and arc itself. 









ML'NSU RATION. 


193 


EXAMPLES. 

1. Required the length of an arc of 22°, in a circle whose 
radius is 5 feet 

Solution. 

Prob. VII, 3.14159x10 = 31.4159 = circumference. 

Arc = 31.4159 x — 1.92, nearly. 

360 ’ J 

2. What is the length of an arc whose chord is 12 feet in a 
circle, whose radius is 14 feet ? 

Solution. 

AC = 14; AD = \AB = = 6. 

From B, (2) B : 14 = sin. A CD : 6. 

Log. i?-flog. 6 = 10.778151 
Log. 14 = 1.146128 

Log. sin. A Cl) = 9.632023 = sin. 25° 22' 3 V 
£ arc = 25° 22' 37" .\ arc = 50° 45' 14" 

Prob. VII., Circumference = 3.14159x28 = 87.96452 
50° 45' 14" = 50.75389°. 

Am. j Arc = 87.96452 x = 12.4015. 

3. What is the length of an arc of 78° in a circle whose 
radius is 16 feet ? 

4. What is the length of an arc whose chord is 20 feet, in 
a circle whose radius is 35 feet. 

PROBLEM XI. 

To find the area of a sector of a-circle. 

Hulk— Multiply the arc of the sector by one half the 
radius. (Geom., B. V, Th. 1.) 

9 





194 


SURVEYING AND N A V i G Af 1 0 N . 




EXAMPLES. 

1. What is the area of a sector of 20°, in a circle whose 
radius is 13 feet? 

Solution. 

Prob. VII, Circumference = 3.14159x26 = 81.68134 

Prob. X, Arc = 81.68134 x = 4.53785 

Sector = 4.53785 x b 3 - = 29.496 sq. it. 

A ns. 29.496 sq. lit. 

2. Required the area of a sector of 32°, whose radius is 20 
feet ? 

3. Required the area of a sector of 18°, whose radius is 1.5 

feet ? Ans. 0.35343 sq. ft. 

PROBLEM* XII. 


To find the area of a segment of a circle. 

Rule.— Determine the area of a sector included between the 
arc of the sector and radii j also the area of a triangle 
formed by the radii with the chord of the segment. If the 
segment be greater than a semicircle take the sum ’ if less , th* 
difference , of these areas: the result will be the area required. 


EXAMPLES. 

1. Required the area of the segment 
AEB , whose arc is 120°, where the radius 
of the circle is 6 feet. 

Solution. 

By Prob. VII. Circumference = 3.14159 x 12 = 37.69908 

By Prob. X. Arc AEB = 37.69908 x = 12.56630 

By Prob. XI. Sector AEBG = 12.56636 x !j = 37.69908 

By Prob. II., R. 2, Trian.A CB = 6x6xsin.l20° = 15 58854 

27 ? __ 

Subtracting, since arc is less than 180°, segment = 22.11054 

Area of segment 22.11054 







MENSURATION. 


195 


\ 


2. Required the area of a circular segment, whose chord is 
8 feet in a circle whose radius is 10 feet ? 

Ans. 4.48 feet, nearly. 

3. What is the area of a circular segment whose chord is 
20 and height 2 feet ? 


PROBLEM XIII. 

To find the area of a zone included between two parallel 
chords. 

Rule.— Take the difference between the areas of the segments 
subtended by the upper and lower bases of the zone, which will 
be the area required. 

If only bases of zone and its 
height are known, the radius of the 
circle may be found as follows: 

Let AB and CD, the bases, 
and EG, the height, be known. 

Draw AC a chord, and from the 
center 0 a perpendicular 01. 

AI = IC. (Geom. B. Ill, Th. 1.) 

Draw also CH parallel to EG, and hence its equal; and IF 
parallel to AG and CE, and hence equal to 

l(AG+CE), 

since I is the middle point of AC. 

In the similar triangles ACII and /6^(Geom.B.II,Th. 1 7, 
Cor. 1), 



CH: AH = IF: FO % 


a) 







196 


SURVEYING AND NAVIGATION. 


Or, EG : AG—CE — AG + CE : EO (2) 

Whence FO = ( 3 ) 

Now, FG — $EG, and GO = FO—FG. Hence, 

an _ AG 2 —CE 2 _ EG _ AG 2 — ( OF 2 + FG 2 ) (£) 

2 FG 2 2 FG K } 

Radius of circle = OB 2 = G0 2 +BG 2 (or AG' 2 ). Sul> 
stituting for GO its value from (4), 

Kadius = OB = p (5) 

V 2 BG / 

Expressed in words, equation (5) will give the following 

RULE. 

To find radius when two parallel chords and their perpen¬ 
dicular distance are given: 

Froyn the square of half the greater chord , subtract the sum 
of the squares of half the lesser chord , and of the height / 
divide the remainder by twice the height y to the square of 
this quotient add the square of one half the greater chord, and 
extract the square root of the whole expression. The result 
will be the value of radius. 

EXAMPLES. 

1. Required the area of a zone whose bases are 80 and 60, 
and their perpendicular distance apart 18.94 feet. 

Solution. 

Taking equation (5) and substituting AG = 40, CF = 30, 
and EG = 1§.94, 

Radius = + = 41 . 

' 37.88 / 

By Prob. VII., circumference = 3.14159 x 82 = 257.61. 














mensuration. 


197 


To find greater segment. 

From (Prob. X., B. Eq. 2), 

12 : 41 = sin.| arc : 40 .*. 
2 arc = 77° 19' 11”. Arc = 154° 38' 22”. 


By Prob. X., Length of arc = 257.61 x = 

360 

“ XI., Sector = 110.65x^.41 = 

“ II., Pule 2, triangle = 

41 x 41 x sin. 154° 38' 22” _ 

2 12 

Greater segment = sector—triangle = 


110.65 

2268.46 

359.996 

1908.464 


To find lesser segment. 

As before, 12 : 41 = sin .2 arc : 30 

l arc = 47° 1' 47” .*. arc = 94° 3' 34”. 

94.0594 


Prob. X., Length of arc = 257.61 x 


360 


“ XI., Sector = 67.31x241 

41 2 x sin. 94° 3' 34” 


“ II., Buie 2, triangle = 
Lesser segment ' 


2 12 


= 67.31 

= 1379.85 
= 838.39 


= 541.46 

Zone=difference of segments=1908.464—541.46= 1367.00. 


2. Kequired the area of a zone whose bases are 96 and 60 
inches, and altitude 26 inches. 

Ajis. 2136.75 sq. inches. 


PROBLEM XI Y. 


To determine the area of an ellipse. 

Bule. —Multiply the product of the semi-axes by 3.14159. 

(For demonstration, see Conic Sections, Ellipse, 16tli 
Theorem.) 








198 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

Required the area of an ellipse whose axes are 12 and 8. 

6x4x3.14159 = 75.40 nearly. 

2. What is the area of an ellipse whose semi-axes are 25 and 
20 feet. 

Ans. 1570.8 sq. ft. 

3. What is the area of an ellipse whose semi-axes are 12 
and 9. 


PROBLEM XV. 

To determine the area of a parabola. 

Rule.— Take two thirds of the product of the base and per¬ 
pendicular height. (For demonstration, see Conic Sections, 
Parabola, Prop. 19th.) 

1. What is the area cf a parabola, the base being 20, and 
the altitude 12. 

* Ans. |.20 x 12 = 160. 

2. What is the area of a parabola when the base is 30, and 
the altitude 20 feet ? 


SECTION III. 

MENSURATION OF SOLIDS. 

In the mensuration of solids, the unit supposed to be applied 
is a cube, receiving its name from the name of one of its edges 
—as, a cubic inch, cubic foot, cubic yard. 

As with surfaces, the ratio of lines is substituted for the 



MENSURATION. 


199 


ratio of solids, and by linear measurements we determine 
solidity. 

The standards for solidity are given in the following 


TABLE OP CUBIC OR SOLID MEASURES. 


1728 cubic inches make 
27 cubic feet 
166 1 cubic yards 
64000 cubic poles 
512 cubic furlongs 


1 cubic foot. 

1 cubic yard. 

1 cubic pole. 

1 cubic furlong. 
1 cubic mile. 


Note.—T he measurement of the surfaces of solid bodies is included in 
this section for convenience. 


PROBLEM I. 

To determine the convex surface of a regular pyramid. 

Rule.— Multiply the perimeter of the base by one half the 

slant height. (Geom., B. YII., Th. 17). 


EXAMPLES. 

1. What is the convex surface of a regular hexagonal pyra¬ 
mid, whose slant height is 12, and each side of its base 5 feet ? 

Ans. 180 sq. ft. 


2. What is the convex surface of a regular octagonal pyra¬ 
mid, whose slant height is 20 feet, and each side of the base 
7 feet ? Ans. 560 sq. ft. 


PROBLEM II. 

To determine the solidity of a py ramid. 


Rule.— Multiply the area of the base by one third of the 
altitude. (Geom., B. YII., Th. 15.) 


200 


SURVEYING AND NAVIGATION 


E X A MPLES. 

1. What is the solidity of an octagonal pyramid, the sides 
of the base being each 8 feet, and the altitude 15 feet? 

Solution . 

By Sec. 2, Prob. YI., Rule 2, 

Area of base = 4.8284271 x 64 = 309.02 

Ana. 309.02 x 5 = 1545.10 cu. ft. 

2. Required the solidity of a pentagonal pyramid, the alti¬ 
tude being 21 feet, and each side of the base 3 feet. 

Ans. 108.39 cu. ft. 


PROBLEM III. 

To determine the convex suiface of a right prism. 

Rule.— Multiply the perimeter of the base by the altitude 
(Geom., B. YII., Th. 3.) 

EXAMPLE. 

* 

1. What is the convex surface of a pentagonal prism, whose 
altitude is 12 feet; and each side of whose base is 2 feet ? 

Ans. 120 sq. ft. 

2. What is the entire surface of a hexagonal prism, having 
an altitude of 7 feet, and each side of whose base is 3.5 feet ? 

Note. —For entire surface, the two bases must be included. 

PROBLEMIV. 

To determine the solidity of a prism. 

Rule. —Multiply the area of the base by the altitude. 
(Geom., B. YII., Th. 11.) 


MENSURATION. 


201 



EXAMPLES. 

1. Required the solidity of the octagonal prism, the altitude 
being 12 feet, and each side of the base 8 feet. 

i 

Solution. 

By Sec. 2, Prob. VI., Rule 2, 

Area of base = 4.8284271 x 64 = 309.02 sq. ft. 

309.02x12 = 3708.24 cu. ft. = solidity. 

2. Required the solidity of an lieptagonal prism, each side 
of the base being 10 feet, and the altitude 30 feet. 

3. Required the solidity of an octagonal prism, whose alti¬ 
tude is 5 feet, and each side of the base 4 inches. 

PROBLEM V. 

To determine the convex surface of a frustum of a regular 
pyramid. 

Rule. — Multiply the sum of the perimeters of the bases by 
one half the slant height of the frustum . (Geom., B. VII., 
Th. 18.) 

/ 

EXAMPLES. 

1. What is the convex surface of a frustum of a regular 
octagonal pyramid; the sides of the bases being 5 and 3 feet 
respectively, and the slant height 6 feet. 

Solution. 

Lower base perimeter 40, 

Upper base perimeter 24, 64 x | = 192 sq. ft. 

2. What is the convex surface of a frustum of a regular 


202 


SURVEYING AND NAVIGATION. 


pentagonal pyramid, the slant height being 12, and the sides 
of the upper and lower bases 5 and 7. 

Note.—I n Problems I. f III., and V., if the entire surface is required, the area 
of the bases must be added to the convex surface. 


PROBLEM VI. 

To determine ike solidity of a frustum of a pyramid. 

Rule.— To the sum of the areas of the bases , add their mean 
•proportional / multiply the sum by one third of the altitude / 
the product will be the required solidity. (Geom., B. VII., 
Tli. 16.) 

EXAMPLES. 

1. Required the solidity of a frustum of a pentagonal pyra¬ 
mid, the side of the bases being 6 and 4 feet, and the altitude 
9 feet. 


Solution. 

Sec. 2, Prop. VI., R. 2, Upper base = 1.7204774 x 16 = 27.53 
Sec. 2, Prob. VI., R. 2, Lower base = 1.7204774 x 36 = 61.94 
Their mean proportional = 1.7204774x24 = 41.29 

Sum = 130.76 

Ans. 130.76 x 3 = 392.28. 

2. Required the solidity of a frustum of an hexagonal pyra¬ 

mid whose altitude is 15 feet, the sides of the bases being 5 
and 2 feet. Ans. 506.65 cu. ft. nearly. 

3. Required the solidity of a frustum of a triangular pyra¬ 

mid whose altitude is 6 feet, the side of the bases bein^ 3 
and 2 feet. Ans. 16.45 cu. ft. nearly. 



MENSURATION. 


203 




PROBLEM YII. 


To determine the solidity of a wedge. 

Definitions. —A wedge is a solid, bounded by five faces, 
viz.: a rectangle, two trapezoids, forming a plane angle, and 
two triangular ends. Tlie common section of the two trape¬ 
zoids is called the edge. 

The base is the rectangular face. 

The altitude is a perpendicular let fall from the edge upon 
the plane of the base. 

Rule.— To the edge add twice the length of the base, and 
multiply the sum by one sixth the product of the altitude and 
breadth of the base . 


Demonstration. 


Let AB — CDEF , be a 
wedge. Through A pass a 
plane parallel to the plane of 
BCD , making AKH , equal 
to BCD. It is then clear 
that the wedge will be di¬ 
vided into two sections ; viz.: 

BCD — AKH , a triangular 
prism, and A—KHEF\ a 
quadrangular pyramid. If 

CF, the length of the base, be longer than AB , the edge, the 
wedge will be equal to the sum of these two sections; if CF 
be the shorter, the wedge will equal their difference. 

Row let MN, the altitude = a , 

AB , the edge ( —ICC) — E r 

CF or ED , length of base — Z, 

EF\ or KI1, or CD, breadth of base = b ; KF will then 
equal L—E. 









204 


SURVEYING AND NAVIGATION. 


It is evident that the prism BCD—AKH is equivalent to 
one half a parallelopiped, whose base is CDHK and altitude 
MN ; hence we have 

Solidity of B CD — AKH— £( CD' KC' ME) — \(b x Ex a) (1) 


- By Problem II., 

Solidity A—KHEF = \ab(L — B) (2) 

But ^{bxExa) = § abE = \abx2>E (3) 

and \ab(L—E) — \abL—\dbE = \abx{fL —2 E) (4) 

Adding values for prism and pyramid as obtained in (3) and 
(4), and we have 

Wedge = \abx{%L — 2E+3E) = ^ab(2L+E) (5) 


Whence the rule. The same result will be obtained when 
E is greater than L. 

EXAMPLES. 

1. Required the solidity of a wedge, when the edge is 9 

feet, its altitude 10 feet, the breadth of the base 6 feet, and 
length of the same 14 feet. Ans. 370 cu. ft. 

2. Required the solidity of a wedge, whose edge is 11 feet, 
its altitude 9 feet, the breadth of base 4, and length 20 feet. 

Ans. 306 cu. ft. 

* 

3. Required the solidity of a wedge, edge being 8, altitude 
12, breadth of base 5, and length 4 feet. 

PROBLEM VIII. 

To determine the surface of a regular polyedron. 

Rule 1 . —Multiply the area of one face by the number of 
faces. 

For, each face is a regular polygon, whose area may be found 
from the length ot one edge; and the faces are also equal. 

Rule 2.— Multiply the surface of a poly edr on of the same 
number of faces , whose edge is unity , by the square of an 
edge of the given poly edr on. (Geom., B. II., Tli. 22.) 


MENSURATION. 


205 


The surface of the polyedron whose edge is unity must be 
obtained by Rule 1. For the convenience of the student in 
the use of Rule 2, we give a table of the surfaces and their 
logarithms. 


TABLE. 


NAME. 

NO. OF 

FACES. 

SURFACE. 

LOGARITHMS. 

Tetraedron. 

Hexaedron. 

Octaedron. 

Dodecaedron. 

Icosaedron. 

4 

6 

8 

12 

20 

1.73205 + 
6.00000 
3.46410 + 
20.64573- 
8.66025 + 

0.2385607 

0.7781513 

0.5395907 

1.3148302 

0.9375307 


EXAMPLES. 

1. What is the surface of an octaedron, each of whose edges 
is 4 inches ? 

Solution. 

(1) By Sec. 2, Prob. VI., Rule 2, 

Area of each face = 0.4330127 x4 1 2 = 6.9282 + 

6.9282 x 8 = 55.4256. 

(2) By Rule 2, 3.46410 x4 2 = 55.4256. Ans. 

2. What is the surface of a dodecaedron, each of whose 
edges is 2.5 inches? 

PROBLEM IX. 

To determine the solidity of a regular polyedron. 

Rule 1 . —Multiply the surface by one third of the perpen¬ 
dicular let fall from the centre upon one of the faces. 




















206 


SURVEYING AND NAVIGATION. 


Demonstration. 

By planes passed through the edges of the polycdron and 
its center, the solid will be divided into a number of pyramids 
whose bases will be the faces of the solid, and whose altitude 
will be the perpendicular from the centre to those faces. Each 
Df these pyramids will be equal (Prob. II.) to its base into 01 3 
third of its altitude; and the sum of the pyramids, that is, the 
polyedron, will equal the sum of the bases, that is, the surface 
into one third the common altitude, which is the perpendicu¬ 
lar from center to face. 

By Buie 1, the solidities of the regular polyedrons having 
unity for each edge have been calculated, and as (Geom., B. VII., 
Th. 19, Cor. 3,) polyedrons are as the cubes of their homo¬ 
logous edges, we have the following: 

Rule 2. Multiply the solidity of a polyedron whose edge is 
unity , and which has the same number of faces, by the cube 
of the edge of the required polyedron . 

For reference we give the solidities of the regular polyedrons, 
as determined by Rule 1, with their logarithms, in the fol¬ 
lowing 

TABLE. 


NAME. 

NO. OF 

FACES. 

SOLIDITY. 

LOGARITHM. 

Tetraedron. 

3 

0.11785 

1.0713344 

Hex aed ron.,<* 

6 

1.00000 

0.0000000 

Octaedron. 

8 

0.47140 

1.6733937 

Dodecaedron. 

12 

7.66312 

0.8844056 

Icosaedron. 

20 

2.18169 

0.3387940 


EXAMPLES. 

1. Required surface and solidity of a regular octaedron, one 
of whose edges is 3 feet. 























Solution. 


Surface = 3.46410 x 9 = 31.1769 
Solidity = 0.47140x27 = 12.7278. 


2. Required the surface and solidity of a regular dodeca* 
edron, each of whose edges is 5 feet. 


( Surface = 516.143 sq. ft. 
t Solidity = 957.89 cub. ft. 


3. Required the surface and solidity of an icosaedron, each 
of whose edges is 7 inches. 


Ans 


( Surface, 424.35 sq. in. 
( Solidity, 748.32 cu. in. 


PROBLEM X. 

To determine the solidity of a prismoid. 

Definition .—A prismoid is a solid bounded by six plane 
faces, two of which are rectangles and parallel, the other four 
being trapezoids. The rectangles are the bases of the figure. 

Rule.— To the areas of the 
bases , add four times the area of 
a section midway between them j 
multiply the sum by one sixth 
the alt itude : the product will be 
the solidity required. 

Demonstration. 

Let ABCD — EFIIK be a 
prismoid. Let a represent the 
altitude; l and b , length and 
breadth of upper base; L and 
length and breadth of lower base. 

If a plane be passed through 
CD — EF, the prismoid will be A 









208 


SURVEYING AND NAVIGATION. 


divided into two wedges, having for common altitude #, 
tiie altitude of the prismoid, and for bases the upper and 
lower bases of the prismoid. 

By Prob. VII., we have the solidity of these wedges respec¬ 
tively : 

Solidity of CD-EFIIK = \ab (2 l+Z) = 

la (2 bl+bL). (1) 

Solidity of EF—ABCD — \aB (2 L + l) = 

\a(2BL + Bl) (2) 

Therefore adding (1) and (2), we have 

Prismoid = \a (2BL + 2 bl + Bl + bL) (3) 

Let m and n be the length and breadth of a section mid¬ 
way between the bases. Then m=\(L + I), and n=l(B + 5), 

mn \^BL-\-bl -}- Bl/-\-bE)^ and 

4 mn — {BL-\~bl-\-Bl-\-bE). (4) 

Substituting from (4) into (3), 4 mn for its equal, we have, 
Prismoid = la(BZ-\-bl-\-4: mn). . (5) 

Whence the rule. 


EXAMPLES. 

1. Required the area of a prismoid, whose bases are 14 by 9 
and 10 by 5, and whose altitude is 18. 

1x18(14x9 + 10x5 + 4(12x7)) = 1536. Ans. 

2. What is the solidity of a prismoid, whose bases are 7 by 5 
and 3 by 3, and whose altitude is 3 feet ? 

Ans. 62 cubic feet. 

3. Required the solidity of a prismoid, whose bases are 17 
by 10 and 5 by 2, and whose altitude is 12 feet. 


MENSURATION. 


I 


209 


PROBLEM XI. 

To determine the convex surface of a cone. 

Rule.— Multiply the circumference of the base by one half 
the slant height. (Geom. B. YII., Tli. 20, Cor. 2.) 

If II — height, we have analytically, 

Convex surface = ixRH. 

EXAMPLES. 

1. Required the convex surface of a cone, the radius of the 
base being 3, and the slant height 11 feet. 

By (Sec. 2, Prob. YII), circumference=3.14159 x 6=18.84 

954 ‘ 18.84954 x! 1 = 103.67. Am. 

2 

2. Required the convex surface of a cone, the radius of 
whose base is 2, and slant height 6 feet. 

3. Required the convex surface of a cone, the radius of 
whose base is 5 inches, and slant height 12 inches. 


PROBLEM XII. 

To determine the solidity of a cone. 

Rule. —Multiply the area of the base by one third of the 
altitude. (Geom. B. YII., Th. 21.) 

Analytically, if A — altitude, we have, 

Solidity = Irr E 2 A. 

EXAMPLES. 

1. Required the solidity of a cone whose altitude is 9 feet, 
and the radius of the base 2 feet. 


1 


210 


SURVEYING AND NAVIGATION. 

Solution . 

By (Sec. 2, Prob. IX., Pule 2), area of base = 3.14159 x 
4 = 12.566+ . 12.566x3 = 37.70. Ans. 

2. Required the solidity of a cone, the radius of whose base 
is 5 feet, and altitude 12 feet. 

3. Required the solidity of a cone, the radius of the base 
being 3 inches, and the altitude 8 inches. 

PROBLEM XIII. 

To determine tJie convex surface of a cylinder. 

Rule. — Multiply the circumference of the hase by the alti¬ 
tude. (Greom., B. VII., Tli. 20, Cor. 1.) 

If A represent the altitude and II the radius of the base, 
we have analytically, 

Convex surface = ^ttRA. 

EXAMPLES. 

1. What is the convex surface of a cylinder, the radius of 
the base being 4, and the altitude 10. 

2t tRA = 2x3.14159x4x10 = 251.3272. Ans. 

2 What is the convex surface of a cylinder whose altitude 
is 5 feet, aod the radius of the base 2 feet? 

3. What is the convex surface of a cylinder whose altitude 
is 3 feet, and the diameter of the base 9 inches ? 

PROBLEM XIV. 

To determine the solidity of a cylinder. 

/ 

Rule. — Multiply the area of the hase hy the altitude. 
(Geom., B. VII., Th. 22, Cor. 1.) 

Exp-essed analytically, Solidity = nR 2 A. 


MENSURATION. 


211 


EXAMPLES. 

1. Required the solidity of a cylinder whose base has a 
radius ot 3 feet, and whose altitude is 10 feet. 

nB 2 A — 3.14159x9x10 = 282.7431 cu. ft. Arts. 

2. Required the solidity of a cylinder whose altitude is 8 
feet, and the radius of the base 5 feet. 

PROBLEM XV. 

To determine the convex surface of the frustum of a cone. 

Rule.— Multiply the sum of the circumferences of the bases 
by one half the slant height of the f rustum. (Geom., B. VII., 
Tli. 20.) 

If B and r represent the radii of upper and lower bases, and 
H the slant height, we have analytical expressions. 

Convex surface = t xll (B + r). 

If the radii of the bases and the altitude of the frustum 
are known, the slant height may be obtained by the following 

Rule.— Add the square of the altitude to the square of the 
difference between the radii , and extract the square root of the 
sum. 

Demonstration. 

Let MJT be the frustum, AB the 
slant height, AD and BC radii of the 
bases, and CD the altitude. Draw BE 
parallel to CD. Then ED — BC and 
AE = AD-BC. Also BE = CD. 

In right-angled triangle AEB , 

AID = BE 2 +AE 2 , 
or substituting and extracting root 

AB = V CD 2 + (AD—BCf 

Whence the rule* 


M 
















212 


SURVEYING AND NAVIGATION. 


EXAMPLES. 

1. Required the convex surface of a frustum of a cone, the 
radii of the bases being 9 and 5 inches, and the altitude 
3 inches. 

Solution. 

Slant height = + (9 — 5)~ = 4^25 = 5. 

By (Sec. 2, Prob. VII.), 

Circumference upper base = 3.14159 x 10 = 31.4159 

Circumference lower base = 3.14159 x 18 = 56.5486 

Sum = 87.9645 

Multiplied by f = 219.911. Ans. 

2. Required the convex surface of a frustum of a cone, the 
radii of whose bases are 2 and 3 feet, and the slant height 
6 feet. 

3. Required the convex surface of a frustum of a cone, the 
radii of whose bases are 4 and 2, and the altitude 3 feet. 

Note.— In Probs. XI., XIII., and XV., to obtain the entire surface, the 
areas of base or bases, must be included. 

PROBLEM XVI. 

To determine the solidity of a f rustum of a cone. 

Rule.— Add the areas of the bases and a mean proportional 
between them ; and multiply the sum by one third the altitude 
of the frustum. (Geom., B. VII., Th. 22.) 

Expressed analytically, A representing altitude, 

Solidity = lAn^R 2 -f r 2 -{-rlt). 

EXAMPLES. 

1. Required the solidity of a frustum of a cone, the radii 
of whose bases are 3 and 4 feet, and whose altitude is 6 feet. 
Solidity = j x 6 x 3.14159 (16 + 9 +12) = 232.48 -. Ans. 




MENSURATION. 


213 

2. Required the solidity of a frustum of a cone, the radii 
ot whose bases are 5 and 7 feet, and altitude 9 feet. 

3. Required the solidity of a frustum of a cone, the radii 
of whose bases are 10 and 13 inches, and whose altitude is 
1 foot. 


PROBLEM XVII. 

To dete7'mine the surface of a sphere. 

Rule. —Multiply the circumference of a great circle by the 
diameter of the sphere. (Geom., B. VII., Th. 25.) 

By (Sec. 2, Prob. VII.), the circumference of a great circle = 
2 nR, or uD. 

Multiply by 2i? or 1 ?, and 

Surface s= 47ri? 2 , or i tJD 2 . 

EXAMPLES. 

1. Required the surface of a sphere whose radius is 5 feet. 

* 

Solution. 

3.14159 x 100 = 314.159 

2. Required the surface of a sphere whose diameter is 2 feet. 

Ans. 12.566. 

3. Required the surface of a sphere whose radius is 11 feet. 


PROBLEM XVIII. 


To determine the surface of a spherical zone. 

Rule. —Multiply the circumference of a great circle by the 
altitude of the zone . (Geom., B. VII., Th. 25, Cor. 1.) 

If A = altitude, we have 

Surface of zone = 2 nR'A 


214 


SURVEYING AND NAVIGATION. 


Remark 1 .— If tlie radii of 
the section forming the zone, be 
known, and the radius ot the 
sphere, the altitude of the zone 
may easily be found. In the 
figure, representing section of 
a sphere, AO and CO — radius 
of sphere, and AK and CII are 
known as radii respectively of 
sections forming the zone. Then 
(Geom., B. I., Th. 39,) VA0 2 -A~K 2 = KO ( 1 ), and 

V CO 2 —CII 2 = no (2). Also RO-KO = IIK , the alti¬ 
tude of zone. 

% 

Remark 2.—If the ares of a great circle subtended by the 
diameters of the sections as chords be known, the altitude 
may also be found. For (in figure above), the arcs CE and 
AE would be known, being halves of the arcs CD and AB. 
Also arc AK = 90°—arc AE. Hence the angles at the 
centre COE and AOK — OAK , would be known. 

Also the angle OCR \ in the K.A. triangle COE — 90° 
-COE. 

Then in the triangles COII&iA AOK , by (Chap. 2, Sec. 2. 
Prop. HI), 

R \ AO — sin. OAK : OK ( 1 ) 

R: CO = sin. OCR : OR (2) 

As before OR— OK = IIK \ the altitude of the zone. 

E X A M P L ES. 

1. Required the surface of a spherical zone on a sphere 
whose radius is 5 feet, and when the radii of the sections are 
4 and 3 feet. 


E 












MENSURATION. 


215 


Solution. 

To find altitude of zone, AO or CO — 5, AK = 4, CH = 3, 
HO = V5 2 —3 2 = 4, KO = V5 2 -4 2 = 3. . v IIK = 1 = M. 

2nBA = 2 x 3.14159 x 5 x 1 — 31.4159 sq. ft. 

2. Required the surface of a zone of a sphere whose radius 
is 9 feet, when the height of the zone is 3 feet. 

3. Required the surface of a zone, on a sphere whose radius 
is 11 feet, where the arcs of the segments whose difference 
forms the zone, are 122° and 58°. 

PROBLEM XIX. 

To determine the solidity of a sphere. 

Rule.— Multiply the surface of the sphere by one third its 
radius. (Georn., B. VII., Th. 29). 

By Prob. XVII., surface — 47 tH 2 or nl) 2 

Multiplying by or \D , Solidity = j7ri? 3 , or 

EXAMPLES. 

1. What is the solidity of a sphere whose radius is 2 inches? 

Ans. 33.51. 

2. What is the solidity of a sphere whose diameter is 40 
i idles ? 

Ans. 33510.4 cu. in. 

3. What is the solidity of a sphere whose circumference is 
24 inches ( 




216 


SURVEYING AND NAVIGATION. 


PROBLEM XX. 

To determine the solidity of a spherical segment. 

Rule. — Multiply the sum of the areas of the bases by one 
half the altitude of the segment , and to the product add the 
solidity of a sphere liavinq this altitude as a diameter. (Geom., 
B. VII., Th. 32.) 

[f R and r represent the radii of bases, and A the altitude 
of the zone, we have 

Solidity of segment = ^(pR 2 -{-nr 2 )-\- \t:A z = 

2 

+ ( 1 ) 

• A 

If the segment have but one base, r — 0, and (1) becomes 

Solidity = —(7? 2 +i^L 2 ) (2) 

2 

EXAMPLES. 

1. Required the solidity of a spherical segment whose bases 
have as radii 10 and 7 inches, and whose altitude is 3.113 
inches. 

ZA{R* + r 2 + fA 2 ) = 3 ' 14159 X A^(lQQ + 49 + 3.23) = 744.6. 

2 2 

Ans. 744.6 cu. in. 

2. Required the solidity of a spherical segment whose bases 
have radii of 5 and 3 feet, and where the radius of the sphere 
is 6 feet. 

Remark. —The altitude of the segment will be found from two right- 
angled triangles, of which radius of the sphere will be hypothenuse, and the 
radii of the bases a side in each. The sum or difference of the third sides of 

4 "' 

these triangles will be the altitude of the zone, according as the sections are 
of different sides, or of the same side of the center of the sphere. 

3. Required the solidity of a spherical segment of one base, 
whose radius is 3 feet, the altitude of the segment being 1.6 
feet. 






MENSURATION. 


217 


PROBLEM XXI, 

To determine the area of a spherical triangle . 

Rule.— Multiply the area of the tri-rectangular triangle , 
or one-eighth of the surface of the sphere , by the excess of the 
angles of the given triangle over two rigid angles , and divide 
the product by 90. (Geoin., Part II., Sec. 1, Prop. 16). 

If A, B, and C\ represent the angles of the given triangle, 
R.A. a right angle, and T the tri-rectangular triangle, we have 

Area = i A + B - + C ~ % 

90 

The division by 90 is necessary in consequence of taking a 
right angle, or 90°, as the unit, instead of 1 degree. (Geom., 
Part II., Sec. 1, Th. 15, Cor. 2.) 

EXAMPLES. 

1. Required the area of a spherical triangle whose angles 
are 62°, 75°, and 102°, on a sphere whose radius is 9 feet. 


Solution. 


Tri-rectangular triangle = }x4 nR 2 = 127.234 = T. 

(A + B+C-2MA.) y _ 62 o + 75 o + 102 o -180 o x m<234 = 


90° 


90 

jj x 127.234 = 83.47. 


Ans. 83.41. 


2. Required the area of a spherical triangle on the same 
sphere, whose angles are 81°, 92°, and 108°. 

3. What is the area of a spherical triangle whose angles are 
70°, 55°, and 87°, on a sphere whose radius is 3 feet ? 

10 






218 


SURVEYING AND NAVIGATION. 


PROBLEM XXII. 

To determine the area of a spherical polygon. 

Rtjle. — From the sum of the angles of the polygon , sub' 
tract twice as many right angles as the figure has sides , less 
two / multiply the remainder by the tri-rectangular triangle , 
and divide hy 90. (Geom., Part II., Sec. 1, Prop. IT). 

If /S' = sum of the angles, and n = the number of sides of 

the polygon, Area = —— ^ j, 

p j 9Q 


EXAMPLES. 


1. Required the area of a spherical polygon of 6 sides, the 
sum of whose angles is 750°, on a sphere whose radius is 6 feet. 
^ surface of sphere = tri-rectangular triangle = 56.56 = T. 
S— 0-2)2 • B.A.\ T _ 750-720 


< 


90 




90 


x 56.56 = 18.85. Ans. 


2. Required the area of a spherical polygon of 5 sides, 
where the sum of the angles is 765, on a sphere having a 
radius of 10 inches. 


RECAPITULATION. 

For convenience, as reference, we give the following resume 
of the expressions for the surface and solidity most commonly 
in use. 

Diameter is represented by D ; radii by R and r ; altitude 
by A ; slant height by li. 

The tri-rectangular triangle is represented by T\ and the 
constant 3.14159 by n. 

Circumference of circle = 2nR or nj) 

Area “ “ = ixR * or 


9 






MENSURATION. 


219 


Arc of circle 

Convex surface of cone 
Entire surface of “ 
Solidity of “ 

Convex surface of cylinder 
Entire “ “ “ 

Solidity “ “ “ 

Convex surface of frustum 
Entire “ u “ 

Solidity “ “ “ 

Surface of sphere 
Solidity of sphere 
Surface of zone 


= 2 

360° 

= 7 xli * h. 

= l ttR 2 * A. 

= 2nl? * A. 

= 2tt^A + 27tA 2 . 

= 7 xR l ' A. 

~ 7r// (R -f- r). 

= nh(Ii + r) -f 7r(A 2 + r 2 ). 

= 477A 2 or ttD 2 . 

= or ' § nD 3 . 

= 27tA * A. 

2 V 3 / 

J. + -5+C'-180°\ 7 , 

90° / 


Solidity of zone (or segment) = 
Area of spherical triangle 





CHAPTER IT. 


LAND SURVEYING. 

SECTION I. 

One of the most important applications of mathematics, for 
men in all departments of life, is Land Surveying. For the 
measurement of the areas of land many processes may he 
given, some exceedingly simple, to be applied to triangular 
and rectangular fields, and corresponding exactly with the 
plain rules of mensuration; others more intricate, and mainly 
interesting as mere mathematical processes. We shall give 
only the more practical, explaining several methods, but dwell¬ 
ing mainly on what is called the rectangular method of com¬ 
puting areas, the method most generally in use, and the one 
best adapted for application in all cases. Before proceeding 
to the rules the attention of the student is called to the defini¬ 
tion of terms as used by surveyors, and to the explanation of 
the traverse table. 


DEFINITIONS. 

The unit of measure for land is the acre, which contains 
10 square chains, or 160 square rods, the surveyor’s chain 
being 4 rods in length. 

The Magnetic Meridian is the direction of the magnetic 
needle, or a North and South line; and a line perpendicular to 
the magnetic meridian is called an East and West line. 


LAND SURVEYING-. 


221 


The Bearing of a Line is the angle it makes with the 
magnetic meridian, and is sometimes called the course. The 
bearing is read from the nearest end of the needle. Thus a 
line that runs so as to make an angle of 40° with the direction 
of the needle, and is on the right hand, is said to be North 
40° East; if it lies on the left hand, it will read North 40° 
W est. 


The Length of a line is the horizontal distance between its 
extreme points. 

The Northing or Southing of a line, or its difference of 
latitude, is the distance between the East and West lines that 
pass through its extremities. 

The Easting or Westing of a line, or its departure, is the 
distance between the meridians passing through its extremities 

If AB represent any distance whose bear¬ 
ing is the angle CAB , then will AC be the 
difference of latitude, and BC the departure 
corresponding to that distance. 

The distance is always the hypotlienuse of 
a right-angled triangle, of which*the latitude 
and departure are the other two sides, and 
the bearing is the angle opposite the depar¬ 
ture. 

Hence, if we multiply the distance by the 
sine of the bearing, the product will be the 
departure. 

And if we multiply the distance by the cosine of the bear* 
ji ig, the product will be the difference of latitude. 



Example 1.—If AB bears N. 22° 30' E., 65.27 chains, what 
is the difference of latitude and the departure ? 

Log. 65.27 = 1.814714 

Cosine 22° 30' = 9.965615 

Log. 60.30 = 1.780329 

Whence AC, the difference of latitude, is 60.30 chains north. 






Log. 65.27 = 1.814714 

Sine 22° 30' == 9.582840 

Log. 24.98 = 1.397554 

Whence BC , the departure, is 24.98 chains east. 

Example 2.—A line bears N. 75° 45' W., 49.50 chains. 
Required the difference of latitude and the departure. 

Log. 49.50 = 1.694605 

Cosine 75° 45' = 9.391206 

Log. 12.18 = 1.085811 

Whence the northing is 12.18 chains. 

Log. 49.50 = 1.694605 

Sine 75° 45' = 9.986427 

Log. 47.98 = 1.681032 

Whence the westing is 47.98 chains. 

If we use the table of natural sines and cosines, we shall get 
the difference of latitude and departure by simple multiplica¬ 
tion, as in the following examples: 


Example 3.—The bearing of a certain line is N. 35° IS' E.; 
distance 12 chains; what is the corresponding latitude and 
departure? 


Angle 35° 18' 
Dis. (multiplier) 
Diff. lat. 


N. cos. .81614 
_12 

9.79368 


N. sin. .57786 

_12 

Dep. 6.93432 


Example 4.—A certain line runs S. 4° 50 f E.; distance 
74.40 ; what is the corresponding latitude and departure ? 


Angle 4° 50' 
Distance 


!N. cos. .9964 
74.4 
39856 
39856 
69748 


N. sin. .0843 
74.4 
3372 
3372 
5901 

Dep. 


Lat. 


74.13216 


6.27192 











LAND SURVEYING. 


223 


In tliis way, by computing for various distances, tlie latitude 
and departure for each degree and quarter degree, or for each 
point and quarter point of the quadrant, and tabulating the 
results, we shall form what is called the 

TRAVERSE TABLE. 

This is a table much used by surveyors and navigators. By 
means ot it, we can, with very little labor, find the latitude 
and departure of any distance and bearing within the limits 
of the table. Thus, in Example 1, if we look under 22° 30', 
we find* 

Latitude for 65 = 60.05; Departure for 65 = 24.87 

Latitude for .27 = .25; Departure for .27 = 10 

Latitude for 65.27 — 60.30; Departure for 65.27 = 24.97 

Whence we have 60.30 chains northing, 
and 24.97 chains easting. 

Example 5.—A line bears S. 43° 30' W., distance 80.25 
chains. Required the difference of latitude and departure. 

In the traverse table, under the angle 43° 30 f , we find, 

Latitude for 80 = 58.03 ; Departure for 80 = 55.07 

Latitude for .25 = .18; Departure for .25 = .17 

Latitude for 80.25 = 58.21; Departure for 80.25 = 55.24 

Whence we have 58.21 chains southing, 
and 55.24 chains westing. 

Example 6.—A line bears 17. 71° 30 f W., distance &5.18, 
chains. Required the latitude and departure. 

In the table, over 71° 30', we find, 

Latitude for 35 =11.11; Departure for 35 = 33.19 

Latitude for .18 = .06 ; Departure for_ .18 = .17 

Latitude for 35.18 = 11.17; Departure for 35.18 = 33.36 

Whence we have 11.17 chains northing, 
and 33.36 chains westing. 













224 


SURVEYING AND NAVIGATION. 


Example 6.—A line bears S. 28° E., distance 155.27 chains. 
Required the latitude and departure. 

In the traverse table, under the angle 2S°, we find 

Latitude for 100 — 88.29; Departure for 100 = 46.95 

Latitude for 55 = 48.56 ; Departure for 55 = 25.82 

Latitude for .27 = .24 ; Departure for .27 = .13 

Latitude for 155.27 = 137.09 ; Departure for 155.27 = 72.90 

Example 7.—A line bears N. 48° 30' W., distance 187.61 

chains. Required the corresponding latitude and departure. 

I ^ j Latitude = 124.31 chains N. 
y ns. j j) e p ar |. ure _ 140 5J u "W. 

Example 8.—A line bears S. 81° W., distance 76.87 chains. 
Required the corresponding latitude and departure. 

j Latitude = 12.02 chains S. 

\ Departure = 75.92 “ W. 

A traverse table computed to two places of decimals 
will answer for the ordinary calculations in land surveying. 
Where greater accuracy is required, and especially where 
the bearing does not agree with any angle in the table, the 
latitude and departure should be obtained by trigonometry, as 
in Examples 1 and 2. 


SECTION II. 

MEASUREMENT OF LINES AND ANGLES, 

1. To measure a line with a chain. 

To measure with a chain requires the assistance of two men; 
a fore chain-man and a hind chain-man. A picket or flag¬ 
staff should be set up in the direction of the line to be meas¬ 
ured. The fore chain-man takes ten pins, and straightens out 


/ 







LAND SURVEYING. 


225 


the chain; the hind chain-man puts his end of the chain at 
the point on the line where the measurement is to begin, and 
by calling right or left , he keeps the fore chain-man in line 
direct toward the flag-staff. When the chain is straight and 
horizontal, the hind chain-man calls down. Then the fore 
chain-man puts into the ground a pin precisely at his end of 
the chain; as soon as the pin is securely fixed, he calls up. 
■Then they advance until the hind chain-man comes to the pin ; 
after seeing that the fore chain-man is in line, the hind chain- 
man brings his end of the chain carefully to the pin, and when 
the chain is straight, he calls down. The fore chain-man then 
puts in another pin at his end of the chain and calls up. The 
hind chain-man then takes up the pin at his end of the chain, 
and they advance in this way until the pins are all down, or 
the end of the line is reached. 

If the pins are all down, a tally must be made, and the ten 
pins again handed to the fore chain-man, when he advances as 
before. This operation must be repeated until the line is 
measured. Great care should be taken that no mistake be 
made in the tally, and that the links are correctly counted at 
the end of the line. 

If the line to be measured is obstructed in any way so as to 
vender its measurement difficult, offsets may be made, and 
pickets set up at equal distances from the line; then the dis¬ 
tance required can be determined by measuring a line in the 
direction of the pickets or flag-staff, and this line will be par¬ 
allel and equal to the line required to be measured. 

2. To find the bearing of a line with the surveyor's compass. 

Place the compass firmly on its support, directly over a 
point in the line; set up a flag-staff at another distant point 
in the line; bring the compass carefully to a level; turn the 
sights accurately to the flag-staff; after the needle has settled 
to its position, read the bearing from the end of the needle 
nearest the direction of the line. If the view along the line 


/ 


226 


i > 

SURVEYING AND NAVIGATION. 

is obstructed so that the flag-staff cannot be seen through the 
sights of the compass, then place the compass at some meas¬ 
ured distance from the line, and set up the flag-staff at the 
same distance from the line, and the bearing of the flag-staff 
will be the bearing required. 


SECTION III. 

MEASUREMENT OF AREAS. 

1. To survey a triangular field. 

Measure the three sides of the 
field with a chain; or measure 
two sides, and with the compass 
. measure the included angle. Then 
compute the area by the rules given 
in Mensuration. 

Example .—Let ABChv a trian¬ 
gular field; measure AB = 24 chains, measure AC — 19.30 
chains, and measure the angle BAC\ equal to 42° 30'. To 
find the area. 

Log. 19.30 = 1.285557 

Log. 24 = 1.380211 

Sin. 42° 30' = 9.829683 

Log. 312.932 = 2.495451 

Whence the double area in chains is 312.932. Divide by 2, 
and we have 156.466, which is the area in chains; move the 
decimal point one place, and we have 15.6466, which is the 
area of the field in acres and decimals of an acre. 


c 






LAND SURVEYING. 


227 


2. To survey a field in the form of a parallelogram. 


Measure two adjacent sides and 
the angle included by them, a 
The area will be found by multi¬ 
plying the two sides, and the sine 
of the included angle. 

Example .—Let A BCD he a 
field in the form of a parallelo¬ 
gram : the side AB was found to 

be 46 chains ; the side AD was found to be 26 chains, and the 
angle at A was 82° To find the area, we have 



c 


Log. 46 = 1.662758 

Log. 26 = 1.414973 

Sin. 82° = 9.995753 

Log. 1184.36 = 3.073484 


Therefore, the area is 1184.36 chains, 

Or, 118.436 acres. Ans . 


Example. ■— Find the 
area from the following 



3. To survey afield where from one corner each of the other 
corners can be seen. 


In the diagram, B,C,D , 
E\ are corners visible from 
the corner A. Measure 
the lines AB,AC,AD and 
AE\ and take their bear¬ 
ings; then in each triangle, 
there wall be two sides and 
the included angle to find 
the area. 


notes: 


I) 






228 SURVEYING AND NAVIGATION. 



BEARINGS. 

CHAINS. 

AB 

W40° E 

15 

AC 

E 

25 

AD 

S 50° E 

30 

AE 

S 20° E 

18 


Whence the angle BAC = 50° ; CAD = 40° ; DAE — 
30°. To find the area of ABC , we have 

Log. 15 = 1.176091 

Log. 25 = 1.397940 

Sin. 50° = 9.884254 

Log. 287.27 = 2.458285 

Whence the double area of ABC is 287.27 chains, which 
gives 14.3635 acres for the area. 

To find the area of CAD , we have 

Log. 25 = 1.3979.40 

Log. 30 = 1.477121 

Sin. 40° = 9.808067 

Log. 482.09 = 2.683128 

Whence the double area of CAD is 482.09 chains, or the 
area is 24.104 acres. 

To find the area of DAE\ we have 

Log. 30 = 1.477121 
Log. 18 = 1.255273 
Sin. 30° = 9.698970 

Log. 270 = 2.431364 

Whence the double area of DAE is 270 chains, or the area 
is 13.50 acres. 

Therefore, the area of ABC — 14.3635 
the area of A CD — 24.1040 
the area of ADE — 13.5000 

51.9675 acres, 

which is the area of the field. 















LAND SURVEYING. 


229 


4. To survey a field from a point within , from which each 
of the corners can be seen . 

Let BCD EE be a field, 
and A a point from which 
all the corners can he seen. 

Measure from the point 
A to each of the corners of 
the field, and also measure 
the angle contained by each 
two lines; then in each tri¬ 
angle there will be two 
sides, and the included angle to find the area. 

Example. —Let AB — 22 chains, 

AG = 15 « 

AB = 20 “ 

AE = 14 “ 

AE •= 18 “ 

Let the angles BAC = 91° 

CAD = 60° 

DAE = 70° 

EAF = 89° 

FAB = 50° 

To find the area of ABC , we have, 

Log. 22 = 1.342423 

Log. 15 = 1.176091 

Sin. 91° = 9.999934 

Log. 329.95 = 2.518448 

Whence ther double area of ABC is 329.95 chains. 

To find the area CAD, we have, 

Log. 15 = 1.176091 

Log. 20 = 1.301030 

Sin. 60° = 9.937531 

Log. 259.81 = 2.414652 

Whence the double area of CAD is 259.81 chains. 






230 


SURVEYING AND NAVIGATION. 


t 

To find the area of DAE\ we have, 

Log. 20 = 1.301030 

Log. 14 = 1.146128 

Sin. 70° = 9.972986 

Log. 263.11 = 2.420144 

Whence the double area of DAE is 263.11 chains. 

To find the area of EAE\ we have, 

Log. 14 = 1.146128 

Log. 18 = 1.255273 

Sin. 89° = 9.999934 

Log. 251.96 = 2.401335 

Whence the double area of EAE is 251.96 chains. 

To find the area of FAB , we have 

Log. 18 = 1.255273 

Log. 22 = 1.342423 

Sin. 50° = 9.884254 

Log. 303.35 = 2.481950 

Whence the double area of FAB is 303.35 chains. 

Therefore the double area of ABC is 329.95 

“ CAD “ 259.81 
“ DAE“ 263.11 
“ EAE “ 251.96 
“ FAB “ 303.35 

2) 1408.18 

The area of BCDEF = 70.409 acres. 

5. To survey afield from two stations within it. 

Measure the distance between the stations. Then measure 
the angle between the line joining the two stations, and the 
line from each station to each corner ol the field. There will 


v 







LAND SURVEYING. 


231 



be then a set of triangles, in eacli of which there will be one 
side and all the angles, 
to determine the other 
sides; after these sides 
are computed, there will 
be a set of triangles about 
each station, in each of 
which there will be 
known the two sides and 
the included angle to 
determine the area; the 
sum of the areas of the triangles of either set will give the area 
of the field. 


Example. —In the diagram let A and B be the stations, 
and C,D,E,F be the corners of the field. Also suppose AB is 
found to be 25 chains, and the angles, 


BAD — 81° 20' 
BAE = 35° 15' 
BAF = 35° 
BAC = 159° 


ABD = 60° 
ABE — 115° 
ABF = 120° 
ABC = 11° 


Then will the angles subtended by the line joining the stations 
be as follows: 


ADB = 38° 40' 
AEB = 29° 45' 
AFB = 25° 0' 
ACB = 10° 0' 


We shall also have the angles, 

DAE = 46° 5'; 
EAF = 70° 15'; 
FAC = 124° O'; 
CAD = 119° 40' • 


DBF = 55° 0' 
EBF = 125° 0' 
FBC = 109° 0' 
CBD = 71° 0' 





232 


SURVEYING AND NAVIGATION. 


In the triangles, 

Log. 25 = 

we have, 

1.397940 

Log. 25 

- 

1.397940 

Sin. 60° 

— 

9.937531 

Sin. 81° 20’ 

— 

9.995013 

Sin. 38° 40' 

- 

11.335471 

9.795733 

Sin. 38° 40' 

— 

11.392953 

9.795733 

Log. AD 

— 

1.539738 

Log. BD 

— 

1.597220 

Log. 25 

— 

1.397940 

Log. 25 

, ~ 

1.397940 

Sin. 115° 

— 

9.957276 

Sin. 35° 15' 

— 

9.761285 

Sin. 29° 45' 

— 

11.355216 

9.695671 

Sin. 29° 45' 

— 

11.159225 

9.695671 

Log. AE 

= 

1.659545 

Log. BE 


1.463554 

Log. 25 

— 

1.397940 

Log. 25 

— 

1.397940 

Sin. 120° 

— 

9.93T531 

Sin. 35° 


9.758591 

Sin. 25° 

—» 

11.335471 

9.625948 

Sin. 25° 

— 

11.156531 

9.625948 

Log. AE 

— 

1.709523 

Log. BE 


1.530583 

Log. 25 

— 

1.397940 

Log. 25 

— 

1.397940 

Sin. 11° 

— 

9.280599 

Sin. 159° 

— 

9.554329 

Sin. 10° 

— 

10.678539 

9.239670 

Sin. 10° 

— 

10.952269 

9.239670 

Log. AC 

—- 

1.438869 

Log. BC 

— 

1.712599 


It is obvious that the area of the field is the same as the 
area of the triangles about each station ; that is, the triangles 
DAE, EAF, FAC ', CAD, about the station A, form the area 
of the field. Also, the triangles DBE, EBF\ FBC\ CBD, 
about the station B form the area of the field. 




















LAND SURVEYING. 


To find the area of the tri¬ 
angle DAE, we have, 

Log. AD = 1.539738 
Log. AE = 1.659545 
Sin. DAE = 9.857543 

Log. 1139.8 = 3.056826 

Therefore 1139.79 is the dou¬ 
ble area DAE. 

For the triangle EAF\ we 
have, 

Log. AE = 1.659545 
Log. ^ = 1.709523 

Sin. 70° 15' = 9.973671 

Log. 2201.6 = 3.342739 

Therefore the double area 
of EAF is 2201.6 chains. 

For the triangle FAC , we 
have, 

Log. AE = 1.709523 
Log. AC = 1.438869 
Sin. FAC = 9.9185 74 

Log. 1166.72 = 3.066966 

Therefore the double area 
of FAC is 1166.72 chains. 

For the triangle CAD , we 
have, 

Log. AC = 1.438869 
Log. = 1.539738 

Sin. 6b4D = 9.938980 
Log. 827.15 = 2.917587 

Therefore the double area 
of CAD is 827.15 chains. 


233 

For the triangle DDE, we 
have, 

Log. BD — 1.59T220 

Log. BE — 1.463554 

Sin. 55° = 9.913365 

Log. 942.19 = 2.974139 

Therefore the double area 
of DBF is 942.19. 

For the triangle EBF\ we 
have, 

Log. BE = 1.463554 

Log. BE = 1.530583 

Sin. 125° = 9.913365 

Log. 808.17 = 2.907502 

Therefore the double area of 
EBF is 808.17 chains. 

For the triangle FBC\ we 
have, 

Log. BF = 1.530583 

Log. BC = 1.712599 

Sin. 109° = 9,975670 

Log. 1655.2 = 3.218852 

Therefore the double area of 
FBC is 1655.21 chains. 

For the triangle CBD , we 
have, 

Log. CB = 1.712599 

Log. BD = 1.597220 

Sin. 71° = 9.975670 

Log. 1929.7 = 3^285489 

Therefore the double area 
of CBD is 1929.7 chains. 











234 


SURVEYING AND NAVIGATION. 


Hence, the triangles about the station A give 

DAE = 1139.79 
EAF = 2201.6 
FAC = 1166.72 
= 827.15 

2) 5335.26 

266.763 acres, the area of the field. 

The triangles about the station B give 

DBE = 942.19 
EBF = 808.17 
FBC = 1655.21 
= 1929.70 

2 ) 5335.27 

266.763 acres, 

the area of the field, which agrees with the above. 

6. To survey a field from two stations without the field . 

Measure the distance 
between the two stations. 

Take the bearing of each 
corner of the field from 
each of the stations, and 
also the bearing of one 
station from the other. 

Then there will be a set 
of triangles in each of 
which there will be 
known one side, and all 
the angles, to determine the other sides—then there will be a 
set of triangles for each station, in which triangles there will 
be known two sides, and the included angle of each, to deter¬ 
mine the area. 











LAND SURVEYING. 


235 


Example. Let CD EF be a field, and A and Ii two sta¬ 
tions without it. Measure A B } and suppose it be found 20 
cliains, and suppose the angles are found as below, 


BA C = 88° 
BAD = 70° 
BAE - 43° 
BAF = 53° 

Then we shall have 

CAD =18° 
DAE= 27° 
EAF = 10° 
FAC — 35° 


ABC = 36° 
ABD = 54° 
ABE = 68° 
ABF = 44° 


CBD = 18° 
DBE = 14° 
ERF = 24° 
i^(7 = 8° 


We have also the angles 

ACB = 56° 
ADB = 56° 
AEB = 69° 
AFB = 83° 


Then to find AC and BC , 

we have 


Log. 20 = 

1.301030 

Log. 20 = 

1.301030 

Sin. 36° = 

9.769219 

Sin. 88° = 

9.999735 


11.070249 


11.300765 

Sin. 56° = 

9.918574 

Sin. 56° = 

9.918574 

Log. AC = 

1.151675 

Log. BC = 

1.382191 


To find AD and BD, we have 


Log. 20 = 1.301030 
Sin. 54° = 9.907958 

11.208988 
Sin. 56° = 9.918574 


Log. 20 = 1.301030 
Sin. 70° = 9.972986 

11.274016 
Sin. 56° = 9.918574 

Log .BD- 1.355442 


Log. AD = 1.290414 










« 


233 SURVEYING AND 

To find AE and BE, we have 

Log. 20 = 1.301030 

Sin. 68° = 9.967166 

11.268196 
Sin. 69° = 9.970152 

Log. AE = 1.298041 

To find AF and BE, we have 

Log. 20 = 1.301030 
Sin. 44° = 9.841771 

11.142801 
Sin. 83° = 9.996751 

Log. AF = 1.146050 

We now have the logarithms c 
angles at each station. 


NAVIGATION. 


Log. 20 = 1.301030 
Sin. 43° = 9.833783 

11.134813 
Sin. 69° = 9.970152 

Log. BE= 1.164661 


Log. 20 = 1.301030 
Sin. 53° = 9.902349 

11.203379 
Sin. 83° = 9.996751 

Log. BF= 1.206628 
the sides of the several tri- 


To find the area of the tri¬ 
angle A CD, we have 

Log. AC =1.151675 
Log. AD = 1.290414 
Sin. CAD = 9.489982 

Log. 85.52 = 1.932071 

Therefore the double area of 
CAD is 85.52 chains. 

To find the area DAE , we 
have, 

Log. AD = 1.290414 
Log. AE = 1.298044 
Sin. 27° = 9.657047 

Log. 176 = 2.245505 

Therefore the double area of 
DAE '\% 176 chains. 


To find the area of CBD, 
we have 

Log. BC = 1.382191 
Log. BD = 1.355442 
Sin. 18° = 9.489982 

Log. 168.89= 2.227615 

Therefore the double area of 
CBD is 168.9 chains. 

To find the area of DBE, 
we have, 

Log. BD = 1.355442 
Log. BE = 1.164661 
Sin. 14° = 9.383675 

Log. 80.13 = 1.903778 

Therefore the double ares of 
DBE is 80.13 chains. 
















LAND SURVEYING. 


23V 


To find the area of EAF \ 
we have, 

Log. AE = 1.298044 
Log. AF — 1.146050 
Sin. 10° = 9.239670 

Log. 48.28 = 1.683764 

Tlierefore the double area of 
FAF \§ 48.28 chains. 

To find the area of FAC , 
we have, 

Log. AF - 1.146050 
Log. AC — 1.151675 
Sin. 35° = 9.758591 

Log. 113.85 = 2.056316 

Therefore the double area of 
FAC is 113.85 chains. 


To find the area of EBF\ 
we have, 

Log. BE = 1.164661 
Log. BF = 1.206628 
Sin. 24° = 9.609313 

Log.95.63 = 1.980602 

Therefore the double area of 
FBF is 95.63 chains. 

To find the area of FBC^ 
we have, 

Log. BF = 1.206628 
Log. BC = 1.382191 
Sin. 8° = 9.143555 

Log. 54 = 1.732374 

Therefore the double area of 
FB C is 54 chains. 


It is obvious that if from the sum of the areas of the trian¬ 
gles CAD and DAE, we subtract the sum of the areas of the 
triangles EAF and FA C, we shall have the area of the field. 
Also, if from the sum of the areas of the triangles CBD and 
DBF, we subtract the sum of the triangles ALSi^and FBC S 
we shall also have the area of the field. 

Therefore, since the double area of 

CAD — 85.52 

DAE = 176.00 

261.52 

EAF = 48.28 

FAC = 113.85 

162.13 

2 ) 99.39 chains. 


Area of the field = 4.9695 acres. 


V 










238 


SURVEYING AND NAVIGATION. 


Again, the double area of 

CBD = 168.89 
I) BE = 80.13 

249.02 

. EBF - 95.63 

FBC = 54.00 

149.63 

2 ) 99.39 chains. 
49.695 acres. 


the area of the field as before. 


SECTION IV. 

1. • . - V ► 

RECTANGULAR SURVEYING. 

1. To survey afield with a chain and compass. 

Begin at any convenient corner of the field to be surveyed, 
•ake the bearing of the first side with the compass, and enter 
it in degrees and minutes in a note book as the first course; 
measure the length of the side with the chain, and enter it in 
chains and decimals, or in chains and links, in the note book 
opposite the first course. Then keeping the field on the right 
hand, proceed to the second side, take its bearing and meas¬ 
ure its length, which measurements enter in the note book ; 
and so proceed until each side of the field has been measured. 
The entries made in the note book are called field-notes. Any 
mark which may distinguish either a side or a corner of the 
field should be carefully entered in its proper place in the 
note book, since it is from these notes that the area must be 
determined, the diagram drawn, and the survey bill written 
out. 







LAND SURVEYING 


239 


2. To find the area of a field whose sides have been meas * 
ured. 

Suppose we have the following field notes: 


1 

N. 23° E. 

17 

chains. 

2 

N. 83° E. 

11 

u 

3 

S. 14° E. 

23 

it 

4 

N. 77° W. 

23.73 

u 


In the diagram let AT CD re¬ 
present the field whose measure¬ 
ments have been taken. Through 
A draw a north and south line 
NS\ from B , C and D draw the 
perpendiculars BE\ CG and BE. 
It is obvious that if from the 
trapezoid CD EG we take the 
triangle DAE\ the triangle ABE 
and the trapezoid FBCG , there 
will remain the field ABCD. 

Now take from the traverse 



table the latitude and departure for each course and distance, 
and arrange all as in the following tablet. 



COURSE. 

DISTANCE. 

_L_. J 

N. 

s. 

E. 

w. 

MERIDIAN 

DIST. 

DOUBLE 

MERID. DIST. 

N. A. 

S. A. 

l B 

N. 23° E. 

17 

15.65 


6.64 


6.64 

6.64 

103.9160 


BC 

N. 83° E. 

11 

1.34 


10.92 


17.56 

24.20 

32.4280 


CD 

CG 

►— > 

o 

tel 

23 


22.32 

5.56 


23.12 

40.68 


907.9776 

DA 

N. 77° W. 

23.73 

5.33 



23.12 

0 

23.12 

123.2296 











259.5736 

907.9776 











259.5736 


Double area in chains = 2) 648.4040 
Area in acres = 32.42020 

















































240 


SURVEYING AND NAVIGATION. 


In tlie tablet we see that AF — 15.65, BH — 1.34, Cl = 
22.32, and EA — 5.33; also FB = 6.64, HC— 10.92, DI — 
5.56, and DE — 23.12. In the right column, the first number 
6.64 is BF; the second number 17.56 is CG , found by adding 
FB and IIC\ the third number 23.12 is DE, found by adding 
CG and DI. The ninth column contains multipliers, found 
by adding the numbers of the eighth column. The first num¬ 
ber 6.64 is FB ; the second number 24.20 is the sum of IB 
and CG ; the third number is the sum of CG and DE ; the 
fourth number is DE. The first number in the tenth column 
is the double area of the triangle ABF\ the second number 
is the double area of the trapezoid FBCG\ the third number 
is the double area of the triangle AED. The number in the 
eleventh column is the double area of the trapezoid CDEG. 

If we subtract the sum of the numbers in the tenth column 
from that of the eleventh, we shall find 648.4040, which is the 
double area of A BCD in square chains. If we divide by 2, 
and move the decimal point one place to the left, we shall get 
32.4202 acres for the area of the field. 

From the above example, we give the following summary 
for finding the area of any field. 

Rule.—1. Prepare a table headed as in the example, 
namely, Bearings, Distance, North, South, East, West, Meri¬ 
dian distance, Double meridian distance, North areas, South 
areas. 

2 . Begin at the most western point of the field, and concei, i 
a meridian to pass through that point. 

Find, by the traverse table or by trigonometry, the northings, 
southings, eastings, and westings of the several sides of the 
field, and set them in the table opposite their respective sta¬ 
tions, under their proper letters, N, S., E., or W. 

m 

3. For the first meridian distance take the departure of the 
first line • for the second , take the first meridian distance and 


LAND SURVEYING. 


241 


add to it the departure of the second line, if the departure is 
east, or subtract if west, dec. 

4. Add each two adjacent meridian distances, and set their 
sum opposite the last of the two in the column of double 
meridian distances. 

5. Multiply each double meridian distance by the latitude 
to which it is opposite, and set the product in the column of 
N. areas if the latitude is north, and in that of S. areas if 
the latitude is south. 


6. Subtract the sum of the N. areas from that of the S. 
areas, and take half the remainder, which will be the area of 
the field in square chains. Dividing this by 10 gives the acres. 


Example .—It is required to find tlie area of the field 
ABCDEF from the following notes. 


G C D 


1 

AB 

N. 20° E. 

= 17.87 chains. 

2 

BC 

N. 30° E. 

= 8.40 “ 

3 

CD 

East. 

= 6.32 “ 

4 

DE 

S. 10° E. 

= 19.20 “ 

5 

EF 

ca 

o 

c 

3 

= 16.80 “ 

6 

FA 

N. 50° W. 

= 12.00 “ 


0 



/ 


K 


Take from the traverse table the 
latitude and departure for each side 
of the field, and arrange them as in ^ 
the following tablet. <g 



















242 SURVEYING AND NAVIGATION. 



BEARING. 

DISTANCE. 

N. 

s. 

E. 

w. 

MERIDIAN 

DIST. 

H 

00 

K ~ 
i-5 P 
w 

p id 
o t: 
p e 

K 

N. A. 

S. A. 

1 

N. 20° E. 

17.87 

16.79 


6.13 


6.13 

6.1o 

102.9227 


2 

3 

• 

w 

o 

o 

8.40 

7.28 


4.20 


10.33 

16.46 

119.8288 


3 

E. 

6.32 



6.32 


16.65 

26.98 



4 

S. 10° E. 

19.20 


18.91 

3.33 


19.98 

36.63 


692.6733 

5 

S. 40° W. 

16.80 


12.87 


10.79 

9.19 

29.17 


375.4179 

6 

N. 50° W. 

12.00 

7.71 



9.19 

0 

9.19 

70.8549 











293.6064 

1068.0912 









• 


293.6064 


774.4348 


38.72534 

Comparing the tablet with the diagram, we find 


Latitudes. 

Departures. 

AO = 16.79 

OB = 6.13 . 

BO = 7.28 

GC = 4.20 

DII = 18.91 

CD — 6.32 

EF = 12.87 

ELL = 3.33 

LA = 7.71 

ELC = 10.79 

FL = 9.19 

Meridian Distances. 

Double Meridian Distances 

BO = 6.13 

BO = C.13 

CP = 10.33 

BO + CP = 16.46 

BP = 16.65 

CP + CD = 26.98 

EM = 19.98 

DP A EM = 36.63 

FL = 9.19 

EM + FL — 29.17 


FL = 9.19 


From the tenth column, we find 

That the double area of AOB = 102.9227 chains, 
“ “ “ OBCP = 119.8288 “ 

“ “ “ ALF = 70.8549 “ 

293.6064 



\ 


\ 







































LAND SURVEYING. 


243 


From the eleventh column, 

Also the double area of PPEM = 092.6733 chains, 

“ “ MEFL — 375.4179 “ 

1068.0912 

Therefore, we have the double area of 

PDEFL = 1068.0912 chains, 

The double area of PGBAFL = 293.6064 

Subtracting, we get 774.4848 

which is the double area of ABCDEF, and dividing by 2 and 
by 10, we get the area 38.72424 acres. 

Example .—Required the area of a field from the following 
notes. 


1. 

N. 33° 30' E. 

= 35.30 

chains, 

2. 

N. 76° E. 

= 16.00 

u 

3. 

S. 

= 9.00 

u 

4. 

S. 10° w. 

= 11.29 

a 

5. 

S. 75° W. 

= 13.70 

u 

6. 

S. 20° 30' W. 

= 10.30 

it 

7. 

w. 

= 16.20 

u 



BEARING. 

DIST. 

N. 

8. 

E. 

w. 

MERIDIAN 

DIST. 

DOUBLE 
| MM RID. DI6T. 

N. A. 

B. A. 

1 

N. 33° 3 O'E. 

35.30 

29.44 


19.49 


19.49 

19.49 

573.7856 


2 

N. 76° E. 

16.00 

3.87 


15.52 


35.01 

54.50 

210.9150 


3 

S. 

9.00 


9.00 



35.01 

70.02 


630.1800 

4 

S. 10° w. 

11.29 


11.12 


1.97 

33.04 

68.05 

-- 

756.7160 

5 

S. 75° W. 

13.70 


3.54 


13.24 

19.80 

52.84 


187.0536 

6 

S. ‘-0° 30'"W. 

10.30 


9.65 


3.60 

16.20 

36.00 


347.4000 

7 

w. 

16.20 




16.20 

0.0 

16.20 












784.7006 

1921.3496 











784.7006 


1136.6490 


Area in acres 56.83245 










































f 


244 


SURVEYING AND NAVIGATION. 


EXAMPLES. 


Required tlie areas of tlie fields wliose sides and bearings are 


given in tlie following tablets. 


( 1 .) 


—'- 

BEARINGS. 

CHAINS. 

1 

s. 

27.00 

2 

N. 87° AY. 

14.00 

3 

N. 4° 13' E. 

40.92 

4 

S. 27°15' E. 

23.00 


Ans. 47 acres. 


(3.) 


— r 

BEARINGS. 

CHAINS. 

1 

S. 80° E. 

3.45 

2 

S. 69° E. 

38.19 

3 

S. 15° 45' AY. 

15.00 

4 

N. 66° 45' W. 

3.15 

5 

N.49°24'W. 

42.26 


Ans . 34.35 acres. 


(5.) 



BEARINGS. 

CHAINS. 

1 

N. 29° 50 1 W. 

10.61 

2 

N. 62° 45'E. 

9.25 

3 

S. 36° E. 

7.60 

4 

S. 45° 30' AY. 

10.40 

_. 


A?is. 8.81 acres. 


( 2 -) 



BEARINGS. 

CHAINS. 

1 

N. 20° 30' E. 

11.66 

2 

S. 79° 45' E. 

20.30 

3 

S. 27° 30' W. 

18.90 

4 

N. 63° 15' W. 

16.56 

5 

N. 15° 30' W. 

2.08 


Ans. 29.S96 acres. 


(4.) 



BEARINGS. 

CHAINS. 

1 

S. 31° AY. 

8.70 

2 

N. 70° 45' AY. 

26.76 

3 

S. 41° 45' W 

6.24 

4 

N. 63° W. 

12.54 

5 

N. 27° 15' E. 

28.26 

6 

S. 49° 26’ E. 

42.30 


Ans. 73.849 acres. 


( 6 .) 



BEARINGS. 

CHAINS. 

1 

N.45° E. 

40.12 

2 

S. 30° AY. 

25.00 

3 

S. 5° E. 

36.00 

4 

AY. 

29.60 

5 

N. 20° E. 

31.00 


Ans. 85.58 acres. 





































LAND SURVEYING. 


?45 


( 8 .) 



BEARINGS. 

CHAINS. 

1 

S.46° 30'E. 

20.00 

2 

S.51°4.y w. 

16.29 

3 

w. 

21.25 

4 

K 56° W. 

27.60 

5 

N. 33°15'E. 

18.80 

6 

S. 77° 13' E. 

32.92 


( 7 .) 



BEARINGS. 

CHAINS. 

1 

K 58° E. 

19.00 

2 

S. 84° E. 

20.00 

3 

S. 17° W. 

20.00 

4 

W. 

20.00 

5 

N.42°19'W. 

15.07 


Ans. 54.95 acres. Ans. 112.90 acres. 

3. To balance the work when the survey is slightly incor¬ 
rect. 

If the field notes have been accurately taken, and the lati¬ 
tudes and departures accurately computed, the sum of the north¬ 
ings will equal the sum of the southings, and the sum of the east¬ 
ings will equal the sum of the westings. This is a good test of 
the accuracy of the work. If the northings do not equal the 
southings, the difference is called the error in latitude; and if 
the eastings do not equal the westings, the difference is called 
the error in departure. When these errors are small they may 
he distributed by the proportion : 

As the sum of the sides of the field is to the error in latitude 
or departure , so is each side to the correction belonging to 
that side. 

In most cases the latitudes or the departures may he made 
equal by taking half the error from the numbers in that col¬ 
umn which gave the greater sum, and adding half the error 
to the numbers in the column which gave the smaller sum ; 
and the area computed from the corrected latitudes and 
departures will be near the truth. 

When any of the sides are more difficult to be measured 
accurately than others, the surveyor must use his judgment in 
applying the corrections. 

















243 


SURVEYING AND NAVIGATION. 


Example .—Required the area of a piece of land, of which 
the following are the field-notes. 



BEARINGS. 

CHAINS. 

1 

K 40° E. 

20.00 

2 

S. 50° E. 

30.00 

3 

S. 40° W. 

40.00 

4 

Kl6j°W. 

36.05 


From the traverse table, we find the following tablet. 



BEARINGS. 

CHAINS. 

LATITUDES 

' 

DEPARTURES. 

N. 

S. 

E. 

w. 

1 

N. 40° E. 

20.00 

15.32 


12.86 


2 

S. 50° E. 

30.00 


19.28 

22.98 


3 

S. 40° W. 

40.00 


30.64 


25.71 

4 

K 16]°IV. 

36.05 

34.61 



10.08 


126.05 

49.93 

49.92 

35.84 

35.19 


The error in the latitudes is one link, and the error in the 
departure is 5 links. As these are small errors, they may be 
so distributed as to make the latitudes and the departures 
balance by the following Rule. 

As the sum of all the sides is to the error in latitude or 
departure , so is each side to the correction for that side , 

Thus, we have the following proportions: 

126.05 : 5 : : 20 : 1, the correction for the first side in 
departure. 

126.05 : 5 : : 30 : 1, for the second side. 

126.05 : 5 : : 40 : 2, “ third side. 

126.05 : 5 : : 36.05 : 1, for the fourth side. 

In this example, the error in latitude being only one link, 




































LAND S U RVE YING. 


247 


it may be applied to tlie southing of the third side, or the 
northing of the fourth side. 

And as the eastings are greater than the westings, the cor¬ 
rection for the first and second sides must be subtracted, and 
those for the third and fourth sides must be added. 

The corrected latitudes and departures are as follows: 



BKAEING8. 

CUAIN8. 

N. 

8. 

E. 

w. 



N. A. 

8. A. 

1 

N. 40° E. 

20.00 

15.32 


12.85 


12.85 

12.85 

196.8620 


2 

S. 50° E. 

30.00 


19.28 

22.97 


35.82 

48.67 


938.3576 

3 

S. 40° W. 

40.00 


30.65 


25.73 

10.09 

45.91 


1407.1415 

4 

N. lepAV. 

36.05 

34.61 



10.09 

0.0 

10.09 

349.2149 











546.0769 

2345.4991 











546.0769 


2 ) 1799.4222 


89.97111 

Whence the area is 89.97 acres. 

It is not always necessary to work out a proportion for each 
correction. The latitudes and departures may be balanced by 
subtracting half the error from the numbers in that column 
which gives the greater sum, and adding half the error to the 
numbers in the lesser column. 

In this method, we usually distribute the half error in pro¬ 
portion to the latitudes or departures. The surveyor should 
also use his judgment in applying the corrections to those 
sides difficult to be measured accurately, so as to make greater 
corrections where errors are most likely to arise. 

Again, the errors may be supposed to arise in part from 
incorrect measurement of the angles, in taking the bearings. 
And it should be borne in mind that when the angle is small, 
the error in its measurement will affect the departure mainly; 
but when the angle is large, the error will be chiefly felt in 
the latitude. 































•248 


SURVEYING AND NAVIGATION. 

> * - V » " 



E X A M P L E S- 

In the following examples correct the latitudes and depar¬ 
tures, and compute the areas. 


(10 ( 2 -) 



BEARINGS. 

CHAINS. 



BEARINGS. 

CHAINS. 

1 

S. 10° w. 

38.00 

# 

1 

K 12° 45' E. 

24.26 

2 

N. 71° W. 

30.50 


2 

S. 71° 55 1 E. 

15.59 

3 

N. 37° 45' E. 

38.00 


3 

S. 34° E. 

9.17 

4 

S. 78° E. 

12.50 


4 

S. 66° 15 W. 

27.69 


Ans. 78.66 acres. Am. 31.31 acres nearly. 


(3.) 


4. To find the hearing and length of a side that has been 
omitted in the survey , and to compute the area. 

If for special reasons the surveyor leaves one side of a field 
unmeasured, the difference between the northings and south¬ 
ings of the measured sides will give the latitude of the un¬ 
measured side, and the difference between the eastings and 
westings of the measured sides will give the departure of the 
unmeasured side; and from these we can get the area of the 



BEARINGS. 

CHAINS. 

1 

N. 20° E. 

15.00 

2 

E. 

10.10 

3 

S. 10° E. 

20.00 

4 

S. 50° W. 

13.50 

5 

N. 30° W. 

16.40 


Ans. 35.42 acres. 


( 4 .) 



BEARINGS. 

CHAINS. 

1 

S. 32° W. 

15.38 

9 

N. 62° W. 

14.26 

3 

N.52° E. 

21.26 

4 

S. 30° E. 

8.20 


Ans. 19.5 acres. 


























V 

LAND SURVEYING. 249 


field; also, the length and bearing of the unmeasured side, as 
in the following example. 



BEARINGS. 

CHAINS. 

1 

S. 40° 30' E. 

31.80 

2 

N. 54° E. 

2.08 

3 

N. 29°15'E. 

2.21 

4 

N. 28° 45' E. 

35.35 

5 

N. 57° W. 

21.10 

6 

• • • • 

• • • • 


Ans. 92.87 acres nearly. 


In the above example the sixth side is unmeasured. To 
find the area, take from the traverse table the latitudes, and 
departures of the five measured sides, and arrange them as 
below. 



BEARINGS. 

DISTANCES. 

N. 

6. 

E. 

w. 

MERIDIAN 

DIST. 

si 

5£ CO 

^ Q 

5 a; 

O 65 

c « 

M a 

N. AREA. 

S. AREA. 

1 

2 

3 

4 

5 

6 

S. 40° 30' E. 
N. 54° E. 
tf.29°15'E. 
N. 28°45'E. 
N. 57° W. 

31.80 

2.08 

2.21 

35.35 

21.10 

1.22 

1.92 

31.00 

11.49 

e » 

24.18 

21.45 

20.65 

1.68 

1.08 

17.00 

17.69 

22.72 

20.65 

22.33 

23.41 

40.41 
22.72 

0.0 

20.65 

42.98 

45.74 

63.82 

63.13 

72.72 

52.4356 

87.8208 

1978.4200 

725.3637 

499.3170 

487.3440 


2844.0401 

986.6610 

986.6610 


2)1857.3791 


92.86895 acres. 


In the tablet we find that the northings exceed the southings 
by 21.45 chains; this we place in the column ot southings, as 
the southing of the unmeasured side. We also find that the 
eastings exceed the westings by 22.72 chains; this we place in 













































250 


SURVEYING AND NAVIGATION. 


tlie column of westings, as the westing of the unmeasured 
side. With these numbers the area is computed as in pre¬ 
ceding examples. To find the bearing of the unmeasured 
side, 

Log. 22.72 = 1.356408 

Log. 21.45 = 1.331427 
Tan. 46° 39' = 10.024981 
whence the bearing is S. 46° 39' W. 

To find the distance 

Log. 22.72 = 1.356408 

Sin. 46° 39' = 9.861638 

Log. 31.24 = 1.494770 
Whence the unmeasured side is 31.24 chains. 

In the following notes the sixth side is unmeasured. He- 
quired the area of the field: 



BEARINGS. 

CHAINS. 

1 

S. 85° W. 

11.60 

2 

N. 53° 30' W. 

11.60 

3 

N. 36° 30' E. 

19.20 

4 

N. 22° E. 

14.00 

5 

S. 76° 30' E. 

12.00 

6 

• • « • 

.3 ( • f 


Ans. 55.08 acres, nearly. 

The bearing of the unmeasured side is S. 13° 18' W., and 
its length is 32.38 chains. 

In the two following examples which are parts of the same 
tract, the 8th side is common, and was not measured. Re¬ 
quired, the area of each part, also the bearing and length of 
the unmeasured side. 











LAND SURVEYING. 


251 


EXAMPLE. EXAMPLE. 


•--— 

BEARINGS. 

CHAINS. 

1 

N. 85° W. 

7.65 

2 

N. 26°j E. 

27.65 

3 

S. 87° E. 

29.99 

4 

S. 3° W. 

26.80 

5 

S. 85° E. 

9.28 

6 

S. 6° W. 

11.61 

7 

8 

N. 85° W. 

• • • • 

19.37 

• * • • 



BEARINGS. 

CHAINS. 

1 

S. 5°| W. 

10.09 

2 

N. 85°| W. 

1.89 

3 

S. 26° W. 

7.20 / 

4 

N. 85°i W. 

18.07 
7.17 ' 

5 

N. 26° E. 

6 

N. 85° W. 

2.97 

7 

N. 5° E. 

21.96 

8 

• • • • 

• • • • 


A?is. 128.985 acres. Ans. 48.975 acres. 

Bearing N. 58° 15 f | W. Bearing S. 58° 15'I E. 

Distance 25.81 chains. Distance 25.81 chains. 

In forming the meridian distance column for the first part, 
begin at the 6th side and add the westings, or begin at the 2d 
side, and add the eastings; for the second part begin at the 
1st side and add the westings. 

5. To draw a plan of a field that has been measured. 

Draw on a paper a line as a meridian passing through the 
first corner of the field, which corner we assume as the start¬ 
ing point; from this, with any scale of equal parts, set off 
along the meridian the northing or southing of the first side of 
the field, as given in the tablet from which the area was com¬ 
puted ; there make an offset equal to the first number in the 
eighth column of the tablet, and it will determine the position 
of the second corner of the field. Then from where the lati¬ 
tude of the first side terminated, measure on the meridian the 
northing or southing of the second side, and make an offset 
equal to the second number in the eighth column, and it will 
determine the position of the third corner of the field. In the 
same manner the other corners can he determined. Lines 
joining these points will represent the sides of the field. 



























252 


SURVEYING AND NAVIGATION. 


Example —It is required to draw a plan from the following 
field-notes. 


0 

BEARINGS. 

CHAINS. 

1 

N. 20° 

E. 

10.00 

2 

S. 80° 

E. 

9.60 

3 

S. 11° 

E. 

10.00 

4 

S. 43° 

W. 

12.00 

5 

N. 31° 

29' W. 

12.73 


Arrange the latitudes and departures as in the tablet below. 



BEARING. 

PI8TANCK. 

* 

N. 

8. 

E. 

w. 

MERID. 

DI8TANCE. 

DOUBLE 
MERID. DI8T. 

1 

N. A. 

8. A. 

1 

N. 20° E. 

10.00 

9.40 


3.42 


3.42 3.42 

32.1480 


2 

S. 80° E. 

9.60 


1.66 

9.45 


12.87 16.29 


27.0414 

3 

S. 11° E. 

10.00 


9.82 

1.91 


14.78 27.65 


271.5230 

4 

S. 43° W. 

12.00 


8.78 


8.13 

6.65 21.43 


188.1554 

5 

N. 31° 29' W. 

12.73 

10.86 



6.65 

0.0 | 6.65 

72.2190 










104.367C 

486.7198 





■ . 


' 



104.3670 









- 



2 ) 382.3528 


Area = 19.11764 acres. 

Draw NS as a meridian, take A as the first corner of the 
field, set off from A to 2 the northing of the first side, 9.40 
chains; at 2 make an offset of 3.42 chains, which will deter¬ 
mine the point B the second corner of the field. From 2 set 
off 1.66 chains, the southing at the second side, and at 3 make 
an offset of 12.87 chains, which will determine the point C 
the third corner of the field ; from 3 set off the southing of the 
third side, 9.82 chains to 4, and at that point make an offset 
of 14.78 chains, and it will determine the point D the fourth 
corner. Then set off* from 4 to 5 the southing of the fourth 
side, 8.78 chains, and at 5 make an offset of 6.65 chains, and 
it will determine the point E the filth corner of the field. 


/ 

















































LAND SURVEYING. - 


Then draw the lines AB , BC\ 
CD , DE and EA , and the plan 
will be completed. 


From the following notes it is 
required to draw a plan of the 
fbld, and compute its area. 



BEARINGS. 

CHAINS. 

1 

N. 20° 30' E. 

10.00 

2 

S. 70° E. 

13.50 

3 

S. 33°30'W. 

20.00 

4 

N. 23° 21' W. 

13.00 


Ans. 17.634 acres nearly. 


2.55 



6. To determine the bearing and distance from one point to 
another , tv hen there are obstructions that render it difficult to 
to take the bearing , or measure the distance directly. 


Begin at one of the points, run on 
several courses in succession, which 
courses must be determined by the 
nature of the ground. Take the bear¬ 
ing and distance of each line run, until 
the other point is reached. The differ¬ 
ence between the sum of the northings 
and the sum of the southings of the 
ines run, will be the northing or 
southing of the line required ; and the 
difference between the sum of the east¬ 
ings and the sum of the westings, will be 
the easting or westing of the required 
line; and from these the line and its 
bearing can be determined. 



i 




















254 SURVEYING AND NAVIGATION. 

Examjyle .—To find the bearing and distance from A to 
v** make the following measurement: 


• ■ - -- 

BEARINGS. 

CHAINS. 

AC 

K 40° W. 

6.30 

CD 

N. 20° W. 

8.50 

DB 

N. 60° E. 

12.20 


Arrange the latitudes as in the tablet below, 



BEARINGS. 

CHAINS. 

N. 

s. 

E. 

w. 

AC 

K. 40° W. 

6.30 

4.83 



4.05 

CD 

K 20° W. 

8.50 

7.99 



2.91 

DB 

N. 60° E. 

12.20 

6.10 


10.56 


AB 




18.92 


3.60 


'Whence we see that 18.92 is the southing from B to A, and 
3.60 is the westing. 

To find the bearing, 

Log. 3.60 = .556303 

Log. 18.92 = 1.276921 

Tan. 10° 46' = 9.279382 

Whence the bearing from B to A is S. 10° 46' W., and from 
A to B, N. 10° 46' E. 

• " - * • * 

To find the distance, 

Log. 3.60 = .556303 

Sin. 10° 46' = 9.271400 

Log. 19.27 = 1.284903 

Whence the distance from A to B is 19.27 chains. 

*; , ' ' - 

_ 

To determine the bearing and distance of a point on the 


V. 




























LAND SURVEYING. 


255 


opposite side of a thicket, a surveyor runs the following lines 
connecting the point with his station, namely: 



BEARINGS. 

CHAINS. 

1 

K 42° E. 

22.40 

2 

N. 14° E. 

21.00 

3 

N. 24° W. 

32.00 

4 

N. 56° W. 

36.00 

5 

S. 50° W. 

16.00 


Required the hearing and distance to the point. 

Ans. N. 24° 44' W = bearing. 
83.80 chains = distance. 


A surveyor running a line north twenty seven degrees west 
meets a thicket so dense that he cannot see his way through 
it; he makes a detour, and takes the following measure¬ 
ment : 


4 

BEARINGS. 

CHAINS 

AC 

N. 20° E. 

12.00 

CD 

N. 10° W. 

15.00 

DE 

1ST. 45° W. 

10.00 

EE 

S. 63° W. 

9.00 


From these notes the surveyor 
wishes to find his line. 

Let AB be a portion of the line 
running N. 27° W., from the bear¬ 
ings and distances, A C, CD , DE\ 
and EF, the bearing and distance 
AF must be computed. 



Arrange the latitudes and departures of these lines as in 
the following tablet: 














/ * 


256 SURVEYING AND NAVIGATION. 



BEARINGS. 

CHAINS. 

N. 

S. 

E. 

-w— 

w. 

AC 

N. 20° E. 

12.00 

11.28 


4.10 


CD 

N. 10° W. 

15.00 

14.77 



2.60 

DE 

N. 45° W. 

10.00 

7.07 



7.07 

EF 

S. 63° W. 

9.00 


4.09 


8.02 

\ FA 




29.03 

13.59 



From this we see that the latitude of FA is 29.03 chains, 
and the departure ot FA is 13.59 chains. Then to find the 


bearing, 

Log. 13.59 

= 1.133219 


Log. 29.03 

= 1.462847 


Tan. 25° 5' 10" 

1 

= 9.670372 


Whence from A to F is N. 25° 5 f 10" W. Subtracting this 
bearing from the bearing of the line AB , which is N. 27° W., 
we have the angle BAF equal 1° 54' 50 ,; . To find AF\ we 
have, 

’ - % 

Log. 13.59 = 1.13321? 

Sin. 25° 5' 10" = 9.627345 

Log. 32.05 = 1.505874 

To find FB, we have, 

Log. 32.05 = 1.505874 

Sin. 1° 54' 50" = 8.523713 

Log. 1.07 = 0.029587 

Therefore the point F is1.07 chains from the line; and as m 
this example the line EF is at right angles to the line that is 
to be found, measure on in the direction BF one chain and 
seven links, and then drive a stake; it will be on the line, and 
from that the line can be continued. To find the distance 
from A to B , we have, 



























LAND SURVEYING. 257 

Log. 32.05 = AF = 1.505874 
Cosine 1° 54' 50' = 9.999778 

Log. 32.04 = 1.505052 

Whence the distance from A to B is 32.04 chains. If the 
surveyor had crossed his line, the bearing of AF would come 
out more than the bearing of the line which he was running; 
that is, in the present example the bearing of AF is not quite 
27°, therefore the surveyor has not quite reached his line at 
the point F 

V 

IRREGULAR BOUNDARIES. 



When the boundary of a field is irregular, it is often advis¬ 
able to run a convenient base line, and from this measure off- 
sets to the prominent points of the irregular boundary. These 
offsets form the parallel sides of trapezoids, and as we measure 
the distance from one offset to another, w r e can compute the 
area included between the base line and the irregular bound¬ 
ary. In the diagram, if ACDEFGB represent an irregular 
boundary of a field, take the bearing and distance from A to 
B\ at C f , B', E', &c., measure offsets C'C, B'B, EE, &c.; 
also measure AO 1 , O'B 1 , BE', &c., from which we can com¬ 
pute the area included between AB and the irregular line. 
If the boundary is curvilinear like the bank of a river, make 
the offsets equally distant from each other, and the area will 
be more easily obtained. 

The area included by the base line will be computed as in 
the preceding cases. 








258 


SURVEYING AND NAVIGATION. 


SECTION Y. 

DIVIDING AND LAYING OUT LAND. 

PROBLEM I. 

To divide a triangle into two parts having a given ratio , b§ 
a line drawn from one of the angles to its opposite side. 

Let ABC represent the triangle; divide 
its base into two parts corresponding to the 
given ratio, and let AD be one of the parts. 

Stake out the line from D to (7, and it will 
divide the triangle as required. 

A D B 

Example 1.—Let AB — 37.50 chains, 

AC — 35.00 “ 

CB = 32.50 “ 

It is required to run the line CD so that the triangle ACD 
shall contain 10 acres. Compute the area of the triangle whose 
sides are given above, it will be found 52.5 acres. Then 

52.5 : 10 : : 37.50 : AD = 7.14 chains. 

Or, without computing the area of the triangle ABC\ we 
may determine AD so that the triangle ACD shall contain 
the required amount of land. 

We have sin . A' AC' AD = %ACD. But in this exam¬ 
ple, the angle A is 53° 8', and the side AC — 35 chains, and 
the triangle ACD is 10 acres or 100 chains. Therefore, we 
have, 

35 sin.53° & • AD = 200* 

Or, AD = -—. =7.14 chains. 

’ 35 sin.53 8' 

Example 2.—A triangular field, whose sides are 20, 18, and 
16 chains, is to have a piece of 4 acres in content fenced off 
from it, by a right line drawn from the most obtuse angle to 


c 










LAND SURVEYING. 


259 


the opposite side. Required the length of the dividing line, 
and its distance from either extremity of the line on which it 
falls. 

Ans, Length of the dividing line, 13 chains 89 links, if run 
nearest the side 16. Distance it strikes the base from 
the next most obtuse angle is 5.85 chains. 


PROBLEM II. 


To divide a triangle into two parts having a given ratio, by 
a line parallel to one of its sides. 

Let ABC be the given triangle, and 
let DE divide the triangle as required, 
then if n : 1 be the ratio of ALE to 
the whole, we must have 

AD : AB :: Vn: 1, 
since similar triangles are to each other as the squares of their 
homologous sides. Therefore, if we take AD, having-to. the 
line AB the ratio of Vn: 1, and from D run a line parallel 
to BC, it will divide the triangle as required. 



Example 1.—In the triangle ABC, if AB — 46 chains, 
AC = 54.75 chains, and BC — 36.8 chains, it is required to 
cut off 20 acres by a line parallel to the side BC. 

The area of the triangle ABC will be found 83.8 acres; then 
we shall have 

AD — 46 f 20 -?=————r = 22.47 chains. 

83-8' (4 . 19) i 

Whence, measure from A 22.47 chains, and then run a line 
with the same bearing as BC\ and the line will cut ofi 20 
acres. 

Example 2.—The three sides of a triangle are 5, 12, and 
13. If two-thirds of this triangle be cut off by a line drawn 
parallel to the longest side, it is required to find the length of 




260 SURVEYING AND NAVIGATION. 


the dividing line, and the distance of its two extremities from 
the extremities of the longest side. 

M / VS - V'2)\ . 

Ans. Distance from the extremity of 5 is 5 ^ / > 

of 12, it is 12 (~ The division line is 13 Vb 

VS ' 


Example 3.—Two sides of a triangle, which include an 
angle of 70°, are 14 and 17. It is required to divide it into 
three equal parts, by lines drawn parallel to its longest side. 
Ans. The first division line on the side 17, cuts that side at 


"\ 


distance 


17 





the second division line, 


17 V2 
VS * 


The 


side 14 is cut at 




14 V2 

Vs 


PROBLEM III 

To divide a triangle into three equal parts by lines dr arm 
f rom the angles to some point within the triangle. 


Let ABC be the triangle whose 
sides are given. Take AD one third 
of the base AB , draw DE parallel to 
AC, and take P the middle of DE\ 
then lines drawn from P to A, B, and 
C will divide the triangle into three 



equal triangles. 

AD being one third of AB, ACD is one third of ACB ; 
but ACP = ACD , since DE is parallel to AC\ therefore, 
A CP is one third of ACB. It is also obvious that CPB and 
APB are equal. The lines AP, BP , and CP are easily 
found, since AD, DP, and the angle at D are known ; whence 
AP will be known, and so of the other lines. Or as the angle 
at D equals the angle at A, we shall have 


AP~ — AD 2 -\-DP~ -f- 2 AD • DP • cos. A , 


which gives AP, since DP is one third of AC. 










LAND SURVEYING. 


261 


Example .—The sides of a triangle are 17.51 chains, 12.575 
chains, and 23.645 chains. This triangle is to be divided into 
three equal parts by lines drawn from the angles to some 
point within the triangle. Required the length of the lines. 

Ans. The line from the angle opposite the side 12.575 is 
13.2242 chains; and the line from the angle opposite 
the side 17.51 is 11.19 chains. 


PROBLEM IV. 

To divide a triangle into two parts having a given ratio , by 
a line running in a given direction . 



Let ABC be the given triangle, DE 
the dividing line, and 1 : n the ratio 
of the triangle ADE to the whole 
triangle. Put AB = c, AC = 5, 

AE — x\ then since the position of a 
DE is given, the angles at D and E 

are known; therefore we shall have AD — -~ n ' and the 

sin. D 

area of the triangle ADE wdll equal - S * n ‘ ^ ; also the 

2 sin. D 

area of ABC will equal - i n ‘ ft Therefore by the con¬ 


ditions we have 


sin .A sin .E‘x 2 


sin. A 'b'c 


2 sin. D 


Whence 


x 


( 


2 % n 

sin. D * b’c 




n sin. E 

which determines the point E\ from which stake out the line 
running in the given direction, and it will divide the triangle 
as required. 

Example .—In the triangle ABC we have AB — 15 chains; 
AC — 12 chains; the angle A = 67°. It is required to 









262 


SURVEYING AND NAVIGATION. 


divide this triangle into two equal parts by a line DE\ which 
makes an angle of 65° with AC, and an angle of 48° with 

AB. 

In this example n = 2, and 

» = ( sin - 65°-12-15 U wa77 cha ; 

\ 2sin. 48° / ’ 

for the line AE, and AD — 8.59 chains. 


PROBLEM V. 

To divide a triangle into two parts having a given ratio , by 
a line passing through a given point. 

Let ABC be the given triangle, 

P the given point; and let DE be 
the dividing line. Put AE = x, 

AD — y, AB = c, AC = b] join 
AP, and put P and P 1 for the perpen¬ 
dicular distance of the given point from AB and AC respec¬ 
tively. Then we shall have 

sm.A’b’c 



ABC = 


2 


ADE = & ' n - A ' x 'y 
2 

Px 


APE = 


2 


APD 


.-P'y 


Then if 1 : n is the ratio of ADE to ABC, we shall have 

( 1 ) 

And since ADE is composed of the two triangles, APE 
and APD, we have 


be 

xy = — 
n 


P'x + P 1 *y = 


sin. A * b • c 


n 


( 2 ) 










LAND SURVEYING. 


263 


Eliminating y from (1) 'and (2), we get 

Sin. A * b * c ,/Sin. 2 A * b 2 * c 2 P'bc'A 

~2Tip ± \ 4 ECP 2 ~EP ) 



When the given point is outside the triangle, the sign 
between the terms within the parenthesis will be positive. 

When the given point is on one of the sides, as on AC, then 
P 1 = 0, and 

sin..A m b'C. 


Again, it is sometimes more convenient to use co-ordinates 
of the given point than perpendiculars ; and if we take the ori¬ 
gin at A, we can measure a line from P parallel to AC, and 
another from P parallel to AB. Call these lines x 1 and y [ ; 

put AE — x. Then will AD = ^ x And the area of 

ryt _ ryt' 

to • 

APE will be y w p ence we s } ia il have, 

2(x—x') ’ ’ 

sin.A*y 'x 2 __ sin.A * b • c 
2(x — x 1 ) 2 n 

ny'x 2 —bcx = — bcx } , 

From which we get, 

_ be ± (bc) 2 (be—^nx'y') 2 

x —----- 

2 ny 1 

Which determines the point E. 

From the above value of x, we see that in some cases there 
may be two lines drawn, each of which will solve the problem. 
When be is less than 4 nx'y 1 , the problem is impossible, or 
when the triangle ADE is less than 4 times the triangle 
formed by the co-ordinates of the given point and the lineAP, 
the problem is impossible : when be — knx ] y\ then x — 2A. 
And then the triangle APE will be the least that can be cut 
off from ABC by a line running through the given point P. 
When the given point is on one of the sides of the given tri- 














264 


SURVEYING AND NAVIGATION. 


angle, as on A C, then will x 1 — 0 , in which case equation (6) 
becomes 


x 


bo_ 

ny r 


Example 1.—In the triangle ABC let 

AB — 30 chains, 

AC = 25 “ 

BC = 20 a 

There is a spring of* water at P ; 
it is required to divide the triangle 
into equal parts by a line running through the spring. 

From the spring measure PF parallel to A C\ suppose it to 
be found equal 4 chains. And measure AF ; suppose it 14 
chains. Then in equation (6), w T e shall have 5 = 25 ; c = 30; 
n = 2; x! = 14 ; y' = 4. 

Therefore, equation (6) gives 

x = 17.13 chains, which is AE. 

And since EF : FP : : AE : AD , we shall find 

AD — 21.89 chains. 

V' • * 

Example 2.—In the triangle ABC\ the angle at A is found 
to be 75° 56', and the co-ordinates of a spring at P are found 
to be AF — 6.40 chains, FP — 5.10 chains. It is required 
to run a line through the spring at P that shall cut off 7 acres 
of land. 

From equation (4), we have, 

Sin.75° 56 1 - 5 .' 10j : ~ = 140, 
x— 6.40 



this being the double area to be cut off in chains. This equa¬ 
tion gives two values of x ; one x — 18.5206, the other x = 
9.7794. The first value, that is AE — 18.5206, will give AD 
== 7.7929 ; the other value of a?, or Ai£=9.7794, gives AD— 
14.7585 chains. Both values are applicable to the problem. 





LAND SURVEYING. 


2J5 


It it had been required to run a line through the spring so 
as to cut off the least quantity of land, we should make FE 
equal to AF, and run the line EPD. The line AE would 
be 12.80 chains, using the numbers in the last example; and 
AD would be 10.20 chains, and the triangle ADE thus cut 
off would contain 6.3322 acres. 

Again, if it were required to cut off 7 acres, and include the^ 
spring by the shortest line DE , we should make AE — AD\ 
then we should have, 

Sin.75° 56' • AE' = 140 chains, 

From which we get, 

AE = 12.014 chains. 

Therefore, if we measure 12.014 chains for AE and AD 
each, the line DE will be the shortest that will cut off 7 acres. 

Example 3.—It is required to find the length and position 
of the shortest possible line which shall *divide into two equal 
parts a triangle whose sides are 25, 24, and 7 respectively. 

Remark.— It is obvious that the division line must cut the sides 25 and 
24 ; and to make it the shortest line possible, the triangle cut off must be 
isosceles. 

Ans. The division line makes an angle with the sides 25 
and 24 of 81° 52 f 10", and its length is 4.899, 


P.ROBLEM VI. 

To divide a quadrilateral into two given parts lyy a line 
passing through a given point in one of the sides . 

Let A BCD be a quadrilateral; it is required to run a line 
from /*, a given point on the side AB , so as to divide the quad¬ 
rilateral into parts such that each shall contain a given area. 
1st. Measure the sides of the quadrilateral, and compute its 

Then extend the two sides as AB and DC until they 

12 


area. 



* 26 b SURVEYING AND NAVIGATION. 
1 ) 



meet in E\ and since the angles B and C of the quadrilateral 
are known, the angles of the triangle BBC are known, and 
the side BC is known; then compute the area of the triangle 
BBC , and also its sides BB and EC. 2d. The point P is 
given in position, therefore BP will be known; add BP to 
BE , and EP will be known. Also, to the area of the triangle 
EBC ', add the area of the part PBCF , and the area of the 
triangl e EPB will be known. 

Then in the triangle EPB we have the area, the angle at 
B , and the side EP , to find the side EE. 

Therefore by mensuration we have 

; EP • EE- sin .E = 2EPF. 


Hence, 


EF = 


2 EPF 
EP • sin .E 


Then subtract EC from EE J and CE will remain. Then 
measure from C towards D the value of CF, and there drive a 
stake. The work will be finished by staking out the line PF. 
If the flag-staff* at A 7 cannot be seen from P, the bearing of the 
line must be determined from the angle EPF\ and the line 
run with the compass. In such cases the surveyor should be 
careful to preserve stakes at equal intervals on the line, so that 
if his bearing does not bring him precisely to the point F\ 
when he reaches the line CD he can measure his distance from 
F\ and by proportion determine how much each stake must 
be moved to mark out a straight line from P to F 




. 







LAND SURVEYING-. 


267 


) 

Example .—Let ABCD be a field with the side AB along a 
highway ; it is required to divide the field into two equal parts 
by a line running from P, a point at the middle of the side 
AB. The following are the field notes: 



BEARINGS. 

CHAINS. 

AD 

N.12°E. 

24.60 

DC 

S. 72° E. 

24.00 

CB 

S. 10° E. 

16.91 

BA 

w. 

30.88 



BEARING. 

DIST. 

N. 

8. 

E. 

w. 



N. A. 

e. a. 

l 

N. 12° E. 

24.60 

24.07 


5.11 


5.11 

5.11 

122.9971 


2 

S. 72° E. 

24.00! 

7.42 

22.83 


27.94 

33.05 


245.2310 

3 

S. 10° E. 

16.91 


16.65 

2.94 


30.88 

58.82 


979.3530 

4 

W. 

30.88 




30.88 

0.0 

30.88 












122.9971 

1224 5840 











122.9971 










- 



2)1101.5869 
Area in acres, 55.07934 


Hence the field contains 55.08 acres; and as it is to be 
divided into equal parts, each part must contain 27.54 acres. 

Produce the sides AB and DC until they meet at E. From 
the bearings of the sides AB and BC, we find the angle 
EBC — 100° ; from the bearing of the sides BC and CD, 
we find the angle ECB = 62° ; therefore the angle at E is 
18°. And since the side BC is 16.91 chains, we have the 
an cries and one side of the triangle EBC to find the other 
sides and the area. 


To find EC, we have . 

Sin. 100° = 9.993351 

Log. 16.91 = 1.2281 44 
= 11.221495 
Sin. 18° = 9.489982 

Log. 53.89 = 

Hence EC = 53.89 chains. 


1.731513 

















































268 


SURVEYING- AND NAVIGATION. 


To (nd EB we have 

Sin. 62° == 9.945935 

Log. 16.91 = 1.2281 44 

11.174079 
Sin. 18° = 9.489982 

Log. 48.317 = 1.684097 

Hence Y B — 48.317 chains. 

Since AB — 30.88 chains, and the point P is at the middle 
of AB, we have PB = 15.44 chains ; therefore EP, the sum 
of EB and BP, will be 63.757 chains. 

To find the area of the triangle, we have 

Log. 53.89 = 1.731513 

Log. 48.317 = 1.684097 
Sin. 18° = 9.489982 

Log. 804.62 = 2.905592 

which is the double area in chains. Hence, the area of the 
triangle is 40,231 acres. 

The area of the part PBCF = 27.54 acres; therefore the 
area of the triangle EPF will be 67.771 acres. Then in the 
triangle EPF, we have the side EP, the angle at E, and the 
area to find the side EF; we shall have 

W _ 1355.42 

63.757 sin. 18°' 

Log. 1355.42 = 3.132074 

Log. 63.757 = 1.804528 
Sin. 18° = 9.489982 

1.294510 1.294510 

Log. 68.796 = 1.837564 

Therefore EF = 68.796 chains. We found EC = 53.89; 
subtracting we get CF — 14.91 chains, which determines 
where the dividing line intersects the side CP; subtracting 
CF from CP, we get FP = 9.09 chains. 









LAND SURVEYING. 


2"0 


To find the bearing and length of the dividing line, we will 
compute the area of one of the parts, say PADF , and thus 
verify the work. 



BEARINGS. 

DIST. 

N. 

S. 

E. 

w. 



N. A. 

S. A. 

PA 

AD 

DF 

w 

N. 12° E. 

S. 72° E. 

15.44 

24.60 

9.09 

24.07 

2.81 

21.26 

5.11 

8.65 

1.68 

15.44 

15.44 

10.33 

1.68 

0.0 

15.44 

25.77 

12.01 

1.68 

620.2839 

33.7481 

35.7168 

/ 

620.2839 

69.4649 

69.4649 


2)550.8190 


which is one half the field. 

To find the bearing of PF\ we have 

Log. 1.68 = 0.225309 

Log. 21.26 = 1.327563 

Tan. 4° 31' = 8.897746 


27.54 acres. 





Hence, .Pi^bears N. 4° 31' W. To find the length of PP,we have 

Log. 1.68 = 0.225309 

Sin. 4° 31' = 8.896246 

Log. 21.33 = 1.329063 
Hence, PF = 21.33 chains. 


PROBLEM VII. —(Gummere’s). 
A piece of land is bounded as follows : 



BEARINGS. 

CHAINS. 

1 

2 

3 

3 

N. 14° W. 
N. 70° 30' E. 

S. 6° E. 
N. 86° 30' W. 

15.20 

20.43 

22.79 

18.00 
















































270 SURVEYING AND NAVIGATION. 

/ 

Within this lot there is a spring; the course to it from the 
second corner is S. 75° E., 7.90 chains. It is required to cut 
off ten acres from the west side of this lot, by a line running 
through the spring. Where will the dividing line meet the 
sides of the lot ? 


Let ABCD be the 
piece of land, and let 
S be the position of 
the spring; extend the o 
lines DA and CB until 
they meet at 0. From the bearings of these lines, we find in 
the triangle AOB , BAO — 72° 30', ABO — 84° 30', and 
AOB = 23°; the side AB = 15.20 chains, AO = 38.72 
chains, ar.d BO = 37.10 chains. The area AOB = 280.67 
chains. Take the origin at O, and consider OG and GS as 
coordinates of the spring. Call OG = x' and GS = y '; and 
since BS = 7.90 chains, and the angle SBL = 34° 30', found 
from the bearing of BS and BC ', we can find OG = x } = 
11.452 chains, and GS = y' = 33.0691 chains. Then put 
x = Oil. LH being the dividing line; and as GS is parallel 
to OL , we shall have 



x—x 1 : y 1 : : x : OI = 


x—od' 


Therefore, we have, 


sin.23° ’y } ’x 2 
%(x—x l ) 


for the area of the triangle OLH\ but the area of the trian¬ 
gle AOB is found to be 280.67 chains, and by the Problem 
the area of ABLII is to be 10 acres or 100 chains. Therefore 
we have, 


sin.23 c 


2(x—x') 


‘JLUl. = 380.67, 


x>_ _ 761.34 

x—x 1 


Or, 


sin.23° * y f 


(i) 








LAND SURVEYING. 


27) 


But, Log. 761.34 = 2.881579 

Log. y' (33.0691) = 1.519422 
Sin. 23° = 9.591878 

1.111300 

Log. 58.922 = 1.770279 

Therefore equation (1) becomes 

■ x ~ , = 58.922 (2) 

X — X 

We liave found — 11.452 chains; this value substituted 
for x 1 in (2) will give, 

a; 2 -58.922 x = -674.77. (3) 

Solving equation (3), we get, 

x = 29.461 ±13.899 

Taking the positive sign, we have, 

x = 43.36 chains, 

Which is the distance from O to H where the dividing line 
meets the side DA. If we subtract OA, which is 38.72 
chains, we have left 4.64 chains for the distance from A to II. 

7 m 

We also find OL == - J/ -~ = 44.94 chains, therefore subtract¬ 
er— xf 

ing ODj which is 37.10 chains, we have BL — 7.84 chains. 

Also the dividing line LH bears S. 2° 58' E., and from L to 
H is 17.67 chains. 

The area of A BCD — 35.6877 acres, and 
“ ABLH =10 acres nearly. 


PROBLEM VIII. 

* 

To cut off a given quantity of land from afield of several 
sides , by a line running from a given point hone of ike sides 
of tke field. 






272 


SURVEYING AND NAVIGATION. 


Let ABCDEF be the field, and 
P the given point on the side AB. 

From P run a random line cutting 
off the required quantity as near as 
may be. Measure PB , BC ', CD, 
and Dh. Compute the area, consid¬ 
ering Ph as the closing side. Also, 
find the bearing and distance of Ph. 

Then if Pa is the true dividing line, 
the triangle Phn will equal the dif¬ 
ference between the area cut off by the random line Ph, and 
the quantity required to be cut off; put this difference equal 
D, then will 

Ph * hn sin.4 = 2 D. 



Therefore 


hn — - 


2D 


Ph * sin .K 


PROBLEM IX. 

To cut off from a field of several sides a given quantity of 
land, by a line running through a given point within the field. 

Let ABCDEF be the proposed 
field, the sides of which should be 
measured in the usual way. Let P 
be the given point; run the line 
IIPh perpendicular to AB at II ; 
compute the area of IIBCDh, and 
find how much it differs from the 
quantity required to be cut off; let 
D equal the difference; let IIP—a, 
and Ph — b. 

Then, if we draw NPn so that the difference of the trian¬ 
gles Pnh and PNII shall equal D, NPn will divide the 
field as required, or will cut oft’ the quantity of land required. 






LAND SURVEYING. 


2'3 


In the triangle PUN, we have PN = —°— h . 

cos. P 

Therefore the area of PUN = = . a . 

2 cos.P 2 cot.P 


In the triangle Pkn , we have P?i = _ ft s * n ‘^ _ 

° sin .(P+h) 

Therefore the area of 


P/m — 


Hence we get 


l 2 


b 2 sm.h sin.P 
2 sin.(P-fA) 2 (cot.P-fcot.A)' 




a* 


cot.P+cot.A cot.P 
And this reduces to 
'a 2 -b 2 


= 2P. 


cot. 2 P + ^ 


2P 


C0t.// j C 


-feot.// ) cot.P = — 


<Pcot.A 

~2TT' 


(2) 

( 3 ) 

W 


From this equation we get the angle P, and thus determine 
the position of the dividing line NPn. 

This Problem can be conveniently solved as follows: 

Extend the sides that are intersected by the dividing line 
until they meet in some point which call 0 ; compute the area 
of the triangle AOE\ and add the quantity to be cut off, and 
take out the area of the triangle AFE\ and we shall have the 
area of the triangle ONn. When NPn is the dividing line, 
let this area equal A ; let xhj be the co-ordinates of the point 
P, the origin being taken at 0 ; these co-ordinates can be 
determined from the position of the point P, the bearings ot 
the sides BA and DE being given. Then put x = On ; then 

will y X - = ON. 
x—x 

Therefore the area of OnN — V x = A. 

2(x—x') 

From the last equation, we can find x, which is the distance 

On\ and since ON = - , we have both On and ON. We 

x—x 
















274 


SURVEYING AND NAVIGATION. 


also have OE and OA ; therefore by subtracting, we have En 
and AN. Hence the position of the dividing line is deter¬ 
mined. 

When the given point is without the field, x' becomes nega¬ 
tive, and we shall have ON = 

x+x 1 


PROBLEM X. 

Let ABCD be a trapezoid • it is required to cut off a given 
quantity of land by a line parallel to AB or CD. 

t 

Produce the lines that are not par¬ 
allel until they meet, compute the sides 
and area of the triangle thus formed 
outside the trapezoid, to the area thus 
formed add the area to be cut off, and 
we shall have the areas of two similar 
triangles and the sides of one of them ; 
and as the areas are as the squares of 
their sides, vve can find the sides of the 
other; and from these we can determine the points ofi 
division. 

Example 1 .— Let ABCD be a trapezoid, AB — 4.50 
chains ; the angle at A is a right angle; the angle at B is 
equal to 106°. 

It is required to cut off six acres of land by a line parallel 
to A B. 

Produce DA and CB until they meet, then will AB tan .B 
— 4.50 tan. 74 = 15.693 chains, which is the distance from 
the point where the sides meet. This will be the altitude of a 
triangle whose base is 4.50 chains, and its area is 35.309 chains; 
to this add the six acres reduced to chains, and ^ve shall ^et 

' O 

95.309 chains for the area of the other triangle; then we can say 



C 







LAND SURVEYING. 


27 J 


35.300 : 05.300 :: (15.693)-: tlie square of the distance from 
the point where the sides meet to the point of division; 
whence we get the distance equal to 25.783 chains. From 
this subtract 15.603 chains, and and we have left AE — 10.09 
chains. 

Therefore measure from A 10.09 chains, and run a line EE 
at right angles to AD, and it will cut off six acres. 

Again, if we put x = the required line AE, a = AE, 

then EF will equal a- f- ——• and hence the area of AEFE 

tan .B 

equals ax-\- - - Therefore putting A for the area to be 

Aj • t (1II* a) 

cut off, we have- + 2ax = 2 A. 

tan .B 

Or, x — — a tan.j5±(2A tan .B + a 2 tan ^B) 1 ’ 

Applying to this equation the value given in the example, we 
get 

x = 10.09 chains, as before. 

Examjyle 2.—There is a farm containing 64 acres ; commen¬ 
cing at its south westerly corner, the first course is N. 15° 
E., distance 12 chains, the second is X. 80° E. (distance lost), 
the third S. (distance lost), the fourth is N. 82° W. (distance 
lost), to the place of beginning. It is required to determine 
the distances lost. 


Note. —Extend the northern and southern boundary westward, and thu3 
form a triangle on the west side of 12. 


Ans. The 2d side is 35.81 
“ 3d “ 23.21 

“ 4th “ 38.76 


chains 

u 


\ J 


u 





SURVEYING AND NAVIGATION. 


PROBLEM XI. 




BEARINGS. 

CHAINS, 

m 

AB 

S. 15°44'W. 

13.80 

BC 

N. 63° W. 

1.40 

CD 

K 25° W. 

9.00 

BA 

K 63° E. 

9.86 


AFbears N. 25° TF, and E is taken on the line of BC distant 
from G 4.60 chains. It is required to run a line from E to 
intersect AF\ so that ABEF shall contain the same quantity 
of land as ABCJD. 



• 











BEARINGS. 

CHAINS 

N. 

8. 

E. 

w. 

5 & 

ei a 

Si 

s 


N. A. 

R. A. 

l 

S. 15° 44' W. 

13.80 


13.28 


3.74 

3.74 

3.74 


49.6672 

2 

N. 63° W. 

1.40 

0.64 



1.25 

4.99 

8.73 

5.5872 


3 

N. 25° W. 

9.00 

8.16 



3.80 

8.79 

13.78 

112.4448 

1 

4 

N. 63° E. 

9.86 

4.48 


8.79 

1 

0.0 

8.79 

39.3792 



157.4112 49.6672 
49.667 2| 


2)107.7440 


Area of ABCD = 53.8720 chains. 


From the bearings of AB and BE we lind the angle 
ABE — 78° 44 f , AB = 13.80 chains, and BE = 6 chains; 


















































LAND SURVEYING. 


277 


thence we find the area of the triangle ABE — 40.G chains. 
This subtracted from 53.872, the area of ABCD , will leave 
13.272 chains, which must be the area of the triangle AEF. 
AVe also find AE — 13.93 chains, and from the bearings of 
AE and AF we get the angle EAF — 114° 17'; and since 
we have 


AF=: 


2AEF 26.544 

AE sin. AEF “ 13.93 sin.TlT^Tf' 


= 2.09 chains. 


Therefore measure 2.09 chains from A to F. Then drive 
a stake, and run a line from E to the stake at F. 

To verify this result compute the area of ABEF\ and com¬ 
pare it with the area of ABCD as before determined. 

When the ground is favorable for staking out parallels, the 
position of the dividing line can be quite easily determined 
in the following manner. 

From the corner D run a line parallel to the diagonal AC, 
and where this parallel intersects the line BCE , drive a stake, 
from which run a line parallel to the diagonal AE, and this 
parallel will intersect the line AF at the point F, and thus 
determine that point; and a line running from E to F will 
be the dividing line required. 


PROBLEM XII. 

In the diagram , AB was 
found to run north 2 chains; 

>BC west 6.40 chains; CD 
was a highway running S. 

40° IF. It was required 
to run a line from A to in¬ 
tersect the highway at some 
point as at D, so as to in¬ 
close 5 acres of land. 



B 






278 


/ 

SURVEYING AND NAVIGATION. 


The area of the triangle ABC is 6.40 chains; this taken 
from the 5 acres leaves the triangle ACD — 43.60 chains. 
AG is found to be 6.71 chains, and the angle ACB — 1 21' 

15"; but the angle BCD — 130°; therefore ACD — 112° 


38' 45". 
Now, 


CD = 


2 A CD 


= 14.09 chains. 


AC sin.ACD 

Then measure along the road 14 chains and 9 links, and 
drive a stake. A line from A to the stake will inclose the 


required quantity of land. 


PROBLEM XIII. 


It is required to divide the field 
ABC DBF into two equal parts, by 
a line running from B, the second 
corner of the field. 

Measure the sides of the field, and 
arrange them as in the following 
tablet. 



1 

BEARINGS. 

CO 

% 

M 

< 

n 

w 

N. 

s. 

E. 

w. 

i 

MERID. 

DISTANCE. 

H 

P 

8 

N. A. 

6. A. 

AB 

N. 15° E. 

14.00 

13.52 


3.62 


3.62 

“ 

3.62 

48.9424 


BO 

N. 20° E. 

20.00 

18.79 


6.84 


10.46 14.08 

264.5632 


CD 

E. 

8.00 



8.00 


18.46 28.92 



DE 

S. 10° E. 

25.00 


24.62 

4.34 


22.80 41.26 


1015.8212 

EF 

S. 15° W. 

15.30 


14.78 


3.96 

18.84 41.64 


615.4392 

FA 

N. 69° 23' W. 
_- 

20.13 

7.09 



18.84 

0.0 

18.84 

133.5756 



447.0812 1631.2604 
! 447.0812 


2 ) 1184.1792 


Therefore the area of the field is 592.0896 chains. 

i 



692.0896 










































LAND SURVEYING. 


279 


Next arrange the latitudes and departures of the sides EF , 
FA, AB, and consider .A A as the closing line, as in the fol¬ 
lowing tablet. 



BEARINGS. 

CHAINS. 

N. 

8. 

E. 

w. 

MERIDIAN 

D1ST. ■ 

MULT. 

N. A. 

EF 

S. 15° W. 

15.30 


14.78 


3.96 

3.96 

3.96 


FA 

N.69°23'W. 

20.13 

7.09 



18.84 22.80 

26.76 

189.7284 

45N. 15° E. 

14.00 

13.52 


3.62 


19.18 

41.98 

567.5696 

BE\ 



5.83 

19.18 


0.0 

19.18 



757.2980 

170.3482 


8. A. 


58.5288 


111.8194 


170.3482 


586.9498 


293.4749 


Whence the area of EFAB is 293.4749 chains; the area 
to be cut off is 296.0448 chains, and the difference is 2.5699 
chains, which is the area of the triangle BEn. 

To find the bearing of BE, we have, 

Log. 19.18 = 1.282849 

Log. 5.83 = .76566 9 

Tan. 73° 5^ = 10.517180 
Whence BE is S. 73° 5*' E. 

To find the distance, we have, 

Log. 19.18 = 1.282849 

Sin. 73° 5J f = 9.980808 
Log. 20.05 = 1.302041 

From the bearing of BE and BE, we find the angle BEB 
equals 63° 5^ f . 

If Bn is the dividing line, we shall have, 


And, 


BE' En sin .BEn 


2.5699, 

5.1398_ 

BE sin. BEn 


2 

En 







































280 


SURVEYING AND NAVIGATION. 


But, Log. 5.1398 = 0.710947 

Sin .BEn or 65° S£ = 9.950234 
Log. BE = 1.302041 

1.252275 

1,252275 

Log. .2875 = 1458672 

Therefore En is nearly 29 links. 

The bearing of Bn is S. 73° 49 f E., and the distance is 
19.92 chains. 


PROBLEM XIV. 


A rectangular farm, 50 chains in length, and 40 chains in 
breadth, containing 200 acres of land, was purchased by tivo 
men, each paying $4,000. The land on one side of the farm 
was found to be worth one dollar per acre more than that on 
the other. The surveyor was required to divide this farm 
equitably between the two 'purchasers, by a line parallel to the 
longest side. 


Let x = the price per acre of the land on one side, then 
#-f 1 will be the price of the other; and will express the 


x 


4000 


number of acres at one price; also, -- will express the num> 

x 1 

her of acres at the other price. These expressions must equal 
the number of acres in the farm ; therefore we have 


— 200 . 

x x-\-l 


This reduces to 


x 2 — 39sc = 20. 


(i) 


(2) 


From which we get 


x 


39.506 nearly. 








LAND SURVEYING. 


2-9! 


Tliis is the price per acre in dollars of the la id on one side 
of the farm, and 40.506 will be the price of the land on the 

other side; and — 101.25 acres, which is the part be- 

oO.oOb 

longing to one purchaser. 


And 


4000 

40506 


= 98.75 acres, which is the part belonging to 


the other purchaser. 


Also, 

And, 


1012.5 
50 

987.5 
_ 50 


= 20.25 chains, the width of one part. 

= 19.75 chains, the width of the other part- 


Therefore, the dividing line will be 19.75 chains from one 
corner, and 20.25 chains from the other. 


PROBLEM XV. 

A rectangular farm , 50 chains in length and 40 chains in 
breadth, is to be divided into two parts of equal value, by a 
line parallel to the longest side , on the supposition that the 
value of the land increases uniformly from one side to the 
other . 


Measuring from the side where the land is of the least value, 
we have 


—= = 28.284 chains to the dividing line: the part on that 
V2 

side will contain 141.42 acres; the other part will contain 
58.58 acres. 


PROBLEM XVI. 

A triangular field, whose base is 35 chains and whose alti¬ 
tude is 20 chains, is to be divided by a line parallel to its base 
into two parts of equal value, on the supposition that the land 






SURVEYING AND NAVIGATION. 


282 


increases uniformly in value from the vertex to the base of the 
trianqle. 

J 20 


Ans. 


The altitude of the triangle will be = 14.142 

^ V2 

chains, and the altitude of the trapezoid will be 5.858 
chains. 


PROBLEM XVII. 


There is a piece of land bounded as follows : Beginning at 
the westernmost point of the field / thence , 



BEARINGS. 

CHAINS. 

1 

N. 35° 15' E. 

23.00 

2 

N. 75° 30' E. 

30.50 

3 

S. 3° 15' E. 

46.49 

4 

N. 66° 15'W. 

49.64 


It is repaired to divide this field into four equal parts , by 
two lines , one running parallel to the third side , the other cut¬ 
ting the first and third sides. 


Find the distance of the parallel line from the first corner 
measured on the fourth side, and the bearing of the other line, 
and its distance from the first corner measured on the first 
side. 


An s. 


Distance to the parallel = 32.50 chains. Bearing 
of the other division line, S. 88° 22' E. Distance, 
5.99 chains. 


TO LOCATE A LINE. 

In running a line that is to be permanent, two flag-staffs 
should be used, so as to allow back-sights to be taken at each 
station. Permanent monuments or marks should be estab- 

s i 

lished at each end of the line, such as a stone, or a stake of some 
durable wood, surrounded by stones. 








LAND SURVEYING. 


283 


Trees directly on the line should be marked as line trees, 
by hewing the bark with an ax on the opposite side of the tree 
in the direction of the line, and then cutting distinct notches 
in the place hewed. . Trees not on the line, but near it, may 
be marked by hewing the bark on the side toward the line in 
two places, more or less distant, according as the tree is nearer 
o T * farther from the line. The corners or points where lines 
intersect should be distinctly marked or indicated by the 
bearing and distance of objects near. 

When the line to be located has two points known, or there 
are two points admitted to be on the line, begin at one of the 
known points, and run on the bearing of the line, if that is 
known ; or if the bearing of the line is not known, run as near 
the true line as may be. Drive stakes at equal distances on 
the line run, as at every 5 or 10 chains, according to the length 
of the line and the condition of the ground. If the line thus 
run does not come to the other given point, measure the per¬ 
pendicular distance from the given point to the line run, and 
by proportion compute the distance each stake is off the true 
line. Then move each stake its corresponding distance, and 
the line required will be marked out by the stakes. 


SECTION YI. 

TRIANGULAR SURVEYING. 

Triangular Surveying is a method of finding the distance 
between remote points by measuring, as a base line, one side 
of the first of a series of triangles, of which each consecutive 
two have a common side, and observing the angles of all the 
triangles in the series. 

Some of the other lines, however, should be measured aftei 



284 


SURVEYING AND NAVIGATION. 


being computed, as a test of the accuracy or inaccuracy of the 
operations. 

Let AB represent a base line, 
which must be very accurately 
measured, for any error in AB 
will cause a proportional error in 
every other line. 

If at A we measure the angles 
BA C, BAD , and at B we measure 
or observe the angles ABC, ABD, 
we then have sufficient data to 
determine the points C and D , and the line CD. 

With the same facility with which we determine the point 
C, we can determine E , or F\ or G , or any other visible 
point. 

Thus we may determine all the sides and angles of the 
iigure CEFGHD , or any visible part of it, by triangulating 
from the base AB. 

The lines forming the triangles are not drawn, except those 
to the points C and D ; we omitted to draw others to avoid 
confusion. 

After any line, as EG , has been computed, it is well to 
measure it, and if the measurement corresponds with compu¬ 
tation, or nearly so, we may have full confidence in the accu¬ 
racy of the work as far as it has been carried. 

We may take CD as the base, and determine any number 
of points visible, as A, B, II , F , G , &c. ; trace any figure 
and determine its area; or show the relative positions and dis¬ 
tances of objects from each other, such as buildings, monu¬ 
ments, trees, &c. 

But to make the computation, triangle after triangle, for the 
sake of making a map, would be very tedious ; and to measure 
every side and angle would be as tedious; and to facilitate this 
kind of operation we may have an instrument called the 

VLANE TABLE. 



♦ 








To determine the distance from A to G, two points remote 
from each other, measure the base AB ; then with a theodolite 
measure the angles of the triangle ABC , the station C 
being selected so as to be visible from A and B ; then com¬ 
pute the sides AC and BC , and measure the angles of the 
triangle BCD, the station D being visible from B and C\ 
then compute the sides BD and CD. Thus continue until 
the point G is reached, when the whole system of triangles 
will be known, and the distance and bearing of the two points 
A and G will be known. 


It sometimes happens that the 
theodolite cannot be placed directly 
over the station, as at C in the dia¬ 
gram ; then it should be placed at P 
as near to C as may be. Measure the 
line PC, and the angles APB and 
BP C ; then 


P 



ABB = APB+PAC, 

Also, ABB = A CB+PBC 

Therefore, we get 

ACB = APB+PAC-PBC. 


From the triangle PAC, we have 

PC&in.APC 


Sin .PAC = 


AC 


from the triangle PBC , we have 

c* ppe PC sin.BPC 

bin. 7 7/ U — - 


BC 










286 


SURVEYING AND NAVIGATION. 


Since PC is small compared with AC, we may take the arc 
for the sine; so also in the triangle PBC. Then substituting 
these values in the above, we «;et 

y O 


ACB = APB+PC I 


sin .A PC sin .BPC\ 


AC 


BC 


) 


AC and BC can be computed from the angle APB, instead 
of ACB, the base AB being known, and the angle APB 
being nearly equal to ACB. The correction above must be re¬ 
duced to arc by dividing by the sine of 1" ; then we shall have 


ACB = APB+ 


PC lAw.APC 
sin. 1"\ AC 


sin. BPC\ 

BC I 


and the correction will be positive or negative according to 
the position of the point P with reference to the station C. 
When the point P is on the circumference of a circle passing 
though the stations A, B, and C, the correction will be 
nothing. 

In this method the stations are selected so as to avoid the 
introduction of either very large or very small angles in any 
of the fundamental triangles, and the line selected for the 
base must admit of being measured with the greatest accuracy. 

After the triangulation lias been carried to considerable 
extent, a line connected with this system, and called a base of 
verification, is selected and measured; the computed length 
compared with the measured length is a test of the accuracy 
of the work. 

In the French Trigonometric Survey under Meehain and 
Delambre, a base of verification was measured and found to 
differ less than one foot from the computed length, depending 
upon a system of triangles extending to the fundamental base 
400 miles distant. 

In the English Survey, General Roy measured a base of 
verification on Romney Marsh, in Kent, and the measured 
length differed only 28 inches from the length computed from 
a system of triangles extending to the fundamental base on 
Hounslow Heath, sixty miles distant. 








LAND SURVEYING. 


287 


In the United States survey, a base was measured on Kent 
Island, in Chesapeake Bay, and another one on Long Island, 
nearly 200 miles distant. The length of one side of a triangle, 
nearly 12 miles, as deduced from one of these bases, differed 
but 20 inches from the length deduced from the other base. 

By computing a system of triangles in this way, we can 
determine the relative positions of places accurately, and give 
great precision to the geography of the country. 

An important application of this method is found in the 
measurement of arcs of the terrestrial meridian, in different 
latitudes, from the results of which we deduce the true figure 
of the earth. 


HARBOR SURVEYING. 

When the triangulation includes coasts and harbors, where 
it is necessary that the depth of water should be indicated, as 
well as the position of rocks, shoals, Ac., signals should be 
anchored on the shoals and rocks, and their bearings from 
each end of a known base must be taken. Then in each 
triangle there will be known all the angles, and one side 
to determine the triangle, and thus the position of the rocks 
and shoals will become known. To indicate the depth of 
water, soundings must be made, and the bearings of signals 
must be taken from the extremities of some known base, 
when, as before, there will be known all the angles and one 
side of each triangle to determine its vertex, which will define 
the position of the sounding. 

Another method is to measure with a sextant the angles 
Included between lines drawn from the place of a sounding to 
three distant objects whose places have been previously deter' 
mined. Then the position of the soundings can be determined 
by Pothenot’s Problem, or the problem of three points. 

In trigonometrical surveying, on shore, the observer is sup¬ 
posed to take his angles from the extremities of a base line; 
but in trigonometrical surveying on water, the observer can 


288 


SURVEYING AND NAVIGATION. 


take liis angles only from single points which may be con¬ 
nected together by distant base lines on the shore. 

Important points along the shore are determined by taking 
latitude and longitude, and intermediate places, by regular 
land surveying. 

The localities of rocks and shoals are also determined bv 

* 

astronomical observations, establishing latitude and longitude, 
in case no land is in sight, or they are far from the shore; but 
in the vicinity of the land, the determination of a point is com¬ 
monly effected by the three point problem. 

The three point problem is the determination of any point 
from observations taken at that point, on three other distant 
points, where the distances of these three points from each 
other are known. 

It is immaterial how those points are situated, provided the 
three points and the observer are not in the same right line, 
the middle one may be nearest or most remote from the ob¬ 
server, or two of them may be in one right line with the 
observer, or all three may be in one right line, provided the 
observer be not in that line. The following example will illus¬ 
trate the principle. 

Coming from sea, at the point D , I observed two headlands, 
A and B , and inland, C, a steeple, which appeared between 
the headlands. I found, from a map, that the headlands were 
5.35 miles from each other; that the distance from A to the 
steeple was 2.8 miles, and from B to the steeple 3.47 miles; 
and I found with a sextant, that the angle ADC was 12° 15' 
and the angle BDC 15° 30'. Required my distance from each 
of the headlands, and from the steeple. 

If the direction of AB is known, the 
direction of AC is equally well known. 

The case in which the three objects, A, C , 
and B , are in one right line, may require 
illustration. 

At the point A , make the angle BAE— 


c 





LAND SURVEYING. 


280 


the observed angle CDB ; and at B, make the angle ABB ■= 
the observed angle ADC. 

Describe a circle about the triangle ABE , join E and C, 
and produce that line to the circumference in D, which is the 
point of observation. Join AD, BD. The angle ADB is 
the sum of the observed angles, and AEB added to it, must 
make 180°. 

The Trigonometrical Analysis. —In the triangle ABE, we 
have the side AB and all the angles; AE and EB can there¬ 
fore be computed. 

In the triangle AEC’, we now have A C, AE, and the angle 
CAE, from which we can compute ACE\ then we know 
A CD. 

Now m the triangle A CD, we have AC and all the angles; 
whence we can find AD and CD. 


SECTION Y 11. 

CANAL AND ROAD SURVEYING. 

Surveys of roads or highways are taken in nearly the same 
manner as the sides of fields are measured. The bearing and 
distance of each portion must be taken, and entered in a note 
book, which will define the position of the main lines; then 
offsets must be made to determine the position of the sides of 
the highway, and the limits of private property. A map must 
be made, on which the principal lines shall be delineated, and 
such permanent monuments referred to, as will enable a sur¬ 
veyor to recover the original lines and limits of the highway, 
should any of them be lost. 



13 



290 


SURVEYING AND NAVIGATION 


CANAL SURVEYING. 

The preliminary surveys of canal routes are conducted in 
the same manner as the survey of roads. A base line is meas¬ 
ured and its bearings taken, from which offsets are made to 
determine the width and position of the canal, and also its 
various embankments. In the re-survey of the New York 
canals, the inner line of the towing path was established 
as the base line. This is called on the maps the red line. 
The outer line of the towing path, determined by offsets from 
the base line, is called the blue line, and this line marks the 
boundary between the property of the state and that of indi¬ 
viduals adjacent to the canals throughout the state. 


PUBLIC LANDS 


Soon after the organization of the present government, sev¬ 
eral of the States ceded to the United States large tracts of 
unoccupied land, and these, with other lands since acquired 
by treaty and purchase, constitute what is called the public 
lands. 

Previous to 1802, there was no general plan for surveying 
the public lands, or in fact, no surveys were made ; and when 
grants were made the titles often conflicted with each other, 
and in some cases different grants covered the same premises. 

In the year 1802, Colonel I. Mansfield, then Surveyor Gen¬ 
eral of the North-western Territory, adopted the following 
method; 

Through the middle, or about the middle of the tract to be 
surveyed, a meridian is to be run, called the principal meri¬ 
dian. At right angles to this, and near the middle of it, 
an east and west line is to be run, and called the principal 
parallel. 

Other meridians are to be run, six miles distant from the 
principal meridian, both east and west. 


< 


\ - * 

LAND SURVEYING. 291 

Also, parallels of latitude are to be run, six miles from the 
principal parallel, both north and south. 

When this is done (and it has been on all the public lands 
east ot the Mississippi river), the whole country is divided into 
squares, six miles on a side, called townships. 

Each township contains 36 square miles. Each square mile 
is called a section, and it contains 640 acres. Sections are 
divided into half sections, quarter sections, and eighths. But 
these divisions are only made on paper. 

Townships which lie along a meridian are called a range , 
and numbered to distinguish them from each other. 

Sections are regularly numbered in every township; and to 
designate any particular one, we say, section 13, in township 
number 4 north, in range 3 east. 

This shows that the third range of townships east of the 
principal meridian, in township No. 4 north of the principal 
parallel, is the township, and the thirteenth section of this 
township is the one sought. 


SURVEY BILL. 


Beginning at a stake and stones, 
thence running north 20° east 
17.87 chains ; thence north 30° east 
8.40 chains, to an elm tree ; thence 
east 6.32 chains to stake and stones; 
thence south 10° east 19.20 chains ; 
thence south 40° west 16.80 chains; 
thence north 50° west 12 chains to 
the place of beginning: contain¬ 
ing by computation 38.72 acres of 
land. 

The survey bill of every piece of 
land measured should be carefully 
written, for the description of the 





v 


292 SURVEYING AND NAVIGATION. 

premises in the title deed is a copy of the Surveyor’s state¬ 
ment. 

Permanent objects near the corners, such as trees, <fcc., 
should be referred to; names of the owners of adjacent lands 
should be stated; also roads and streams of water should be 
so mentioned as clearly to define the premises. 


SECTION VIII. 

VARIATION OF THE COMPASS. 

As the true meridian is an astronomical line, we must find 
it by astronomical observations; and then by comparing the 
meridian of the compass with it, we shall have the variation 
of the compass. 

When the sun is on the equator, it rises due east, and sets 
directly in the west. Should we then observe the direction of 
its center, just as it was rising or setting, at the time it had no 
declination, and trace that line a short distance on the ground, 
we should have a due east and west line. 

If from any point in that line we draw another line at right 
angles, we should then have the true meridian. 

If we now put the compass on this meridian, and make the 
sight-vanes range with it, the needle will also range with it, 
if there is no variation. But if the north point of the needle is 
to the west of the sight-vane, the variation is westerly; if to 
the east, easterly ; and the number of degrees and parts of a 
degree that the needle deviates from the direction of the sight- 
vanes shows the amount of the variation. 

But it is not to be supposed that any particular observer 
can be at the points and places, where the sun is either rising 
or setting just at the time the sun is on the equator. We 



LAND SURVEYING. 


293 


must have a broader basis, and in fact by means of the lati¬ 
tude of tlie observer and the declination of the sun, any 
observer has the means of knowing the precise direction in 
which the sun will rise or set, any day in any year. 

Let us suppose that the sun on a certain day, observed from 
a certain place, must have risen S. 81° E.,but by the compass 
it was observed to rise S. 79° E., the variation of the compass 
was therefore 2° west. 

These observations are called taking an azimuth. Azimuths 
are often taken at sea to determine the variation of the 
compass. 

On land, however, the horizon is rarely visible and very 
few observations on sunrise or sunset can be made ; besides 
there are other objections arising from atmospherical refrac¬ 
tion. It is therefore best, most convenient, and more condu¬ 
cive to accuracy, to take the sun when up 10°, 15°, or 25° above 
the horizon, observe its direction per compass, and compare 
the result with the computed bearing for the same moment, 
and if the two results agree the compass has no variation ; if 
they disagree the amount of such disagreement is the amount 
of the variation of the compass. 

The true bearing of the sun can be determined when the oh 
server knows his latitude, the sun’s declination, and the alti 
tude of the sun. 

In the diagram let Zj he the 
zenith of the observer, P the 
north pole, and S the place of 
the sun when its altitude was 
^aken. Then will PZ be the 
observer’s co-latitude ; PS will 
be the sun’s co-declination, and 
ZS will be the sun’s co-altitude, 
and the angle PZS will give 
the sun’s bearing or azimuth. 









294 


SURVEYING AND NAVIGATION. 

In the spherical triangle PZS , the three sides will be known, 
and to find an angle, we have the equation, 

Cos .PS — cos .PZ cos.SZ+sin.PZ sin.SZ cos .Z (1) 

If we put A = the sun’s declination, Z = the observer’s 
latitude, and A — the sun’s altitude; then since A is the 
complement of PS, Z that of PZ , and A that of SZ, the 
above equations will become 

Sin.A = sin.Z sin.H + cos.Z cos.H cos.Z. (2) 

from which we can determine the angle Z by its cosine. 

Example. 

In latitude 39° 6' 20” north the sun’s declination was 
12° 3' 10” north, and the true altitude of the sun’s center was 
observed to be 30° 10' 40”. What was the sun’s azimuth? 

In this example A — 12° 3' 10”, Z — 39° O' 20”, and A = 
30° 10' 40”; with these values we get PZS — 99° 17', or 
II ZS — 80° 43'. Hence the bearing of the sun is S. 80° 43' E. 

If at the time of taking the altitude of the sun another 
observer had taken its bearing by the compass, and found it to 
be S. 80° 43' E., then the compass would have no variation, 
and whatever it differed from that would be the amount of 
variation. 

If a line were run along the ground, direct toward the 
center of the sun, at the time the altitude was taken, and 
sufficiently marked, that would be a standing line of known 
direction; and if from any point in that line, we could 
draw another line, making an angle with it of 99° 17' on the 
north, or 80° 43' on the south, such a line definitely marked, 
would be a permanent meridian line for all time to come, on 
which we could at any time place a compass, and observe its 
variation. 


LAND SURVEYING-. 


295 


Instead of equations (1) and (2) we may use the equation 


\ 


Cos. = /sin./S'si n.^- 

1 • T \ rj 


sin PZ sin./_ 



from which we get Zhy the cosine of half Z , where S is half 
the sum of the sides. 

Here PZ — 50° 53' 40"; PS — 77° 50' 50"; ZS — 59° 49' 20" 
And S- 94° 19' 55", and S-PS = 16° 23' 5". 


Hence Sin.A = +9.998758 


&m.(S—PS) = +9.450381 

Sin .PZ = —9.8S9853 

Sin.ZS' = —9.936750 


2)19.622536 


Cos. 49° 3S'30' = 9.811268 

Therefore Z = 99° 17' 12" nearly as before. 

This equation is adapted to logarithms; see Spherical Trig- 
onometry. 

TO LOCATE A MERIDIAN LINE FROM OBSERVATIONS UPON 
THE NORTH STAR WITH A THEODOLITE. 

The north star, Polaris, is now 1863, about 1° 25 ; distant 
from the north pole, and the star apparently makes a circle 
round the pole in a sidereal day, making two transits across 
the meridian, one above and the other below the pole—a 
direction to it, at these times, would be a true meridian line. 

To find these times, subtract tlce right ascension of the sun 
from the right ascension of the star ; increasing the latter by 
24h., to render the subtraction possible, when necessary. 

The difference will be the time of the upper transit, and 
llh. and 59m. from that time will be the time of the 
lower transit. The right ascension of the sun is to be found 






296 


SURVEYING AND NAVIGATION. 


in the Nautical Almanacs, for every day in the year; and it 
is nearly the same, for the same day, in every year. 

For example. At what time will the north star make its 
transits over the meridian on the first day of July, 1853. 

IT. M. S. 

* i?. A. -f 24h. 25 6 0 

O P. A ... 6 41 16 

18 24 44 

This result shows that the upper transit will occur about 
6h. 24m., in the morning of the 2d of July. I say about , 
because I took the sun’s right ascension for the morning of 
July 1, and from that time to 6, next morning, is 18 hours; 
and during this time the right ascension of the sun will 
increase full 3 minutes—therefore the upper transit will take 
place 6h. 21m. in the morning, and the previous lower transit 
llh. 59m. previous, or at 6h. 22m., evening. 

The time when the north star is on the meridian may be 
known approximately, since the star in the handle of the 
dipper nearest the four stars that form the dipper, passes the 
meridian nearly at the same time as the north star. Having 
obtained the time the pole star will be on the meridian, direct 
the telescope so that at that moment the star shall be upon 
the middle spider line, and the line of the telescope will indi¬ 
cate the true meridian ; and this line permanently marked will 
enable the surveyor to ascertain the variation of his compass 
at any time. 

Or knowing his latitude, and having from the Almanac the 
north star’s polar distance, and its time of passing the meri¬ 
dian, the surveyor can find when the pole star will be at its 
greatest elongation east or west from the equation 

Cos.P = tan.Ztan.A, 

Where Z is the latitude of the place, and A is the north star 
polar distance, and P is the time angle, which angle reduced 





297 


LAND SURVEYING. 

to time by allowing 15° for one hour, will give the time of the 
stai s greatest elongation. This may be approximately known 
by the position of the dipper ;Uhe first star in the handle of 
the dipper will be in a horizontal line with the north star. 
Diieet the telescope to the star a little before this time, and 
follow the star until it ceases to move in the same direction. 
Then mark the direction of the telescope. 

Compute the star’s azimuth from the equation 

sin.A = cos.Z * sin.Z, 

Where A is the star’s polar distance, Z is the latitude, and 
Z the angle of azimuth. 

Then with the theodolite upon the line of greatest elonga¬ 
tion, set ofi the angle Z as determined from the above equa¬ 
tion, and the line thus run will be the true meridian. 


Example, 

In 1863, the polar distance of the north star is about 1° 25k 
Required its azimuth for latitude 43° 3k 
From the last equation, we have, 


Sin.Z 


sin.A 
cos.Z’ 


And since A = 1° 25', and Z = 43° 3', we have, 

Sin. 1° 25' = 8.393101 

Cos. 43° 3' = 9.863774 


Sin. 1° 56' 20" = 8.529327 

Which is the angle Z, or the azimuth of the pole star at that 
time. If the direction of the star at the time of its greatest 
elongation has been marked out, then with the theodolite 
mark out another line intersecting the first at an angle of 
1° 56' 20", and it will be the meridian. 

By placing a compass on any well defined and true meridian, 
we can determine its variation by simple observation. 





298 SURVEYING AND NAVIGATION. 

The declination of the needle, or the variation, as it is 
called, is the angle between the true and the magnetic meridian. 

In the United States, the magnetic needle points west of 
north at all places in the eastern and middle states, and east 
of north in the western states. 

Those places where the true and magnetic meridian coincide, 
or where the needle points directly north, are said to be on the 
line of no declination ; and those places where the angle be¬ 
tween the true and magnetic meridian is the same, are said to 
be on lines of equal declination. 

At the close of the last century, the Western declination in 
the United States was decreasing, and the eastern declination 
was increasing; this is now reversed, the western declination 
is increasing, and the eastern is decreasing. In the Kew 
England States, the western declination is increasing from 5 1 
to 7 f annually. In the middle states, the western declination 
is increasing at the rate of from 3' to 4' annually; and in the 
southern and western states at the rate of about 2 f annually. 

The change of declination is not constant at the same place, 
and at different places is quite unequal. In 1835, Charlottes¬ 
ville, in Virginia, was on the line of no declination, and the 
line of no declination then struck Lake Erie, not far from Erie 
in Pennsylvania. In 1840, the line of no declination was west 
of Charlottesville, so the declination of that place was 19 f west 
while the line struck Lake Erie, not far east of Cleveland, in 
Ohio, the declination of that place being then 19 f east. 

In the 34th and 39th volumes of the American Journal of 
Science, Professor Loomis published charts with lines of equal 
declination for the greater part of the LTnited States, arranged 
for the years 1838 and 1840. 

In the Coast Survey Report for 1856, under the direction 
of Dr. Bache, a chart with lines of equal magnetic declination 
was published. This chart embraced the Pacific Coast, and 
the lines were arranged for the year 1850. 

- The following table shows the magnetic declination of seve¬ 
ral places in the United States, 

* ' ■ ‘ . ' - • * 


% 


LAND SURVEYING. 299 


TABLE OF DECLINATIONS. 


PLACE. 

DECLINATION. 

• DATE. 

Maine. 

. . N. E. angle. . 

19° 12' W. 

r* 

1838 

Cambridge. 

.. Mass. 

9° 18' W. 

1840 

Burlington. 

..Vt. 

9° 45' W. 

1837 

New Haven. 

..Ct. 

6° 10' W. 

1845 

Columbia College .. 

N Y 

• • A 1 • A. « « • ••• « « 

6° 25' W. 

1845 

Albany. 

N Y 

• • -A 1 • -M- ••••• • • • 

7° 54' W. 

1855 

Washington. 

p> p 

• • -L' • V_y • ••••••• 

5° 44' W. 

1855 

Buffalo. 

..N. Y. 

1° 25’ W. 

1837 

Philadelphia. 

. . Pa. 

4° 8'W. 

1840 

Pittsburg. 

..Pa. 

0° 33' W. 

1845 

Charlottesville. 

.. Ya. 

0° 19' W. 

1840 

Hudson. 

. . Ohio. 

0° 52' E. 

1840 

Baleigh. 

NT P 

• • i • V— * • • • • •••• 

0° 44' E. 

1854 

Charleston. 

..s. c. 

2° 30' E. 

1849 

Cincinnati . 

..0. 

4° 26 1 E. 

1840 

Detroit . 

.. Mich . 

2° 00' E. 

1840 

Mobile . 

..Ala . 

7° 05' E. 

1850 

Nashville . 

. . Ten.-. . 

7° 07' E. 

1835 

Alton . 

..Ill . 

7° 45' E. 

1840 

St. Louis . 

.. Mo . 

8° 37' E. 

1840 

Natchez . 

.. Miss . 

9° 00' E. 

1802 

San Francisco . 

..Cal . 

15° 27' E. 

1852 


The change of declination can be ascertained from the 
recorded bearings of old lines, as the division lines of farms. 
Place the compass upon the line whose bearing is given, and 
direct the sights to a flag-staff upon another distant point of 
the same line ; the reading of the compass compared with the 
recorded bearing of the line will give the change of declina¬ 
tion. 

To retrace lines from the hearings given in “ ancient 
deeds.” First, ascertain the change of declination since the 
dates of the record; then if the change has been west, add the 



















































300 


SURVEYING AND NAVIGATION. 


change to the S. W. and N. E. bearings, and subtract the 
change from the N. W. and S. E. bearings ; if the change has 
been east, then subtract the change from the S. W. and N. E. 
bearings, and add it to the N. W. and S. E. bearings. The 
bearings thus corrected will give the included angle at each 
corner of the field the same as the old bearings. 


Example. 

In an old deed the following bearings w^ere given : 



BEARINGS. 

1 

S. 10° w. 

2 

K 71° W. 

3 

N.37° E. 

4 

S. 53° E. 


The change of declination w^as found to be 3° 30' west. 
What will be the bearings of the sides ? 



BEARINGS. 

1 

S. 13°30'W. 

2 

N. 67° 30' W. 

3 

N. 40° 3<V E. 

4 

S. 49° 30' E. 


If at the time the above bearings were taken the declination 
of the needle were 3° 30' west, what would be the bearings 
of sides referred to the true meridian? 



BEARINGS. 

1 

S. 6° 30' W. 

2 

N. 74° 30' W. 

3 

N. 33° 30' E. 

4 

S. 56° 30' E. 


A ns. 















LAND SURVEYING. 


30! 


The 1 ines of an old survey may be retraced by moving the 
the vernier of the compass through an arc equal to the change 
in the declination of the needle since the old survey was made ; 
then with the compass thus adjusted, run the lines according 
to the bearing given in the notes of the old survey. 

Or place the compass on a well marked line of the old sur¬ 
vey, and move the vernier until the needle indicates the same 
hearing as given for that line in the notes of the old survey ; 
then the compass will retrace the lines of the survey from the 
original bearings, and the reading of the vernier will indicate 
the change of declination since the original survey was made. 

Any land mark to the corner of a lot laid down by the 
original surveyors, must remain ; subsequent surveyors can 
straighten lines between point and point, and decide what the 
true courses are, and how many acres the lot contains. 

When a surveyor is called to survey any farm or estate that 
has been previously surveyed, he must find some corner as a 
place of commencing, and from this run a random line, as 
near the true line as his judgment permits ; and if he strikes 
another corner he has run the true course; if not, he corrects 
his course, as taught in Chapter IT. Thus he must go round 
the field from corner to corner. He has a right to establish 
corners only where no corners are to be found, and no evidence 
can be obtained as to the existence and locality of a former 
land mark. 

It may be the case that a surveyor is called to survey a lot 
where no corners are to be found. If a fence or line exists, 
which has been the undisputed boundary for a long time, that 
boundary cannot be changed, and the surveyor must establish 
a corner by ranging some other line to meet the first. Some¬ 
times corners may be found to some neighboring lot, from 
which lines can be run to establish a corner to the lot we wish 
to survey. 

Lines of lots in the same town are generally parallel; and a 
surveyor wdio offers his services to the public, must make him¬ 
self acquainted with the general directions of the lines of 


/ 


302 SURVEYING AND NAVIGATION. 

lots, over that section of country where his services are 
required. 

When a surveyor is called to divide a piece of land, lie is 
then an original surveyor, and not liable to be embarrassed by 
old lines and old traditions ; he has then only his mathematical 
problem before him. 

Owing to the inaccuracies of original surveys, and the im¬ 
possibility of leaving proper land marks, in consequence of 
the great haste in which lands were originally surveyed, great 
confusion has followed, in some sections of our country, in 
respect to lines, and it has been no uncommon thing to have 
whole neighborhoods at variance, if not in law, in reference to 
the boundaries of their lands. 




•' . ‘ J 








> 





















CHAPTER V. 


TOPOGRAPHICAL SURVEYING. 


SECTION I. 
LOCATING CURVES. 


PROBLEM I. 

* To stake out a curve with the Compass or Transit . 



Suppose the line TA is tangent to the curve at A and 
TO is tangent at C. Measure A B and determine the angle at 
B ; then will r , the radius of the required curve, he equal to 
AB tan. and the angle BAP will be found from the 

equation — = sin .BAP, where c is the engineer’s chain; 
2 /* 


then with the transit at A, make an angle BAP as deter¬ 
mined by the above equation, and from A measure one chain to 
P, where put a pin ; make the angle BAP' double the angle 
BAP , and measure one chain from P to range with AP } ; 
put a pin at P . Make the angle BAP 11 three times the angle 
BAP ; measure from P 1 one chain in range with AP\ and put 




304 


SURVEYING AND NAVIGATION. 


a pin at P u , and so on ; the pins at P , P 1 , P n , &c., will be in 

the circumference of a circle. 

» 

If it is not convenient to measure AP, take the bearing and 
distance from A to C; then will the radius of the required 


l AC 


- = r, whence we can get the angle PAP 


curve be _ _ 

sin.PAC 

as before. When AB — PC, the lines are tangents at A and 
C, but otherwise not. By the last method the curve must pass 
through the given point C, but may not be so that the given 
line will be tangent at the point C ; when the point C is not 
given, we may assume any convenient radius, and then stake 
out the curve. 


PROBLEM II. 

To locate a curve with a chain. 

Let r be the radius of the required curve, and c the length 

o 

of the chain in feet; then will _ be the versed sine of the arc 

2 r 

whose chord is the given chain. 



Suppose the straight line AP is tangent to the curve at A. 
Put a pin at A, and extend the chain its length from the pin 
along the line AP ; deflect the chain into the position AB', so 

that the versed sine ~ shall measure the distance of P } from 

the line AP. Put a pin at B', extend the chain along the 

line AP ' to C; then deflect the chain to C, so that CC’ shall 

2 

be double the versed sineL_, or double the deflection at B' 

2 r 

Put a pin at 6", then extend the chain along the line PC 1 to 










TOPOGRAPHICAL SURVEYING. 3Q5 


D , and deflect tc D f making tlie deflection DC 1 equal to CC, 
and so on. 

The pins at A, B r , C 1 and D', will be in the circumference 
of a circle, and AB', B'C', CD’, Ac., will ho e<pial chords in 
a circle whose radios is r. 

When the curve is again to unite with a straight line as at 


D r , the next deflection, as at E, must he half that at D 1 , or 

the same as at the first point B' ; that is, the deflection must 
c 2 

be_, the given versed sine. 

2 r 


If,' for example, the radius of the required curve is 1,000 


feet, the chain being 100 feet, then — will equal 5 feet, the 

2 /* 


deflection at B' ; at C 1 , the deflection will be 10 feet, while 
at E' the deflection will be 5 feet, and D'E' will be tangent 
to the curve at E l . 


PROBLEM III. 


To stake out a parabola to which two given lines shall be 
tangents. 

Let AE and ED ' be the 
given lines meeting at E. 

Measure off equal spaces, as 
ED, DC, &c., on the line 
EA ; put stakes at the points 
A, B, C, D, &c. Also meas¬ 
ure off equal spaces on the 
line ED\ and put stakes as 
at A', B’, O', &c. 

Range out AA 1 and BB l , 
and put a stake at the inter¬ 
section of the ranges at P\ then range out CC', and put a 
stake at P where the range intersects that of BB ] ; range out 
DD\ and at P n put a stake where DD ' intersects CC The 







306 


SURVEYING AND NAVIGATION. 

stakes at P, P, P u , will be on the arc of a parabola, and the 
line AE will be tangent at ^l,and ED 1 will be tangent at D l ; 
this method will apply whether AE = ED S or not. 


SECTION II. 
LEVELING. 


The surface of tranquil water is called a level surface; a line 
whose points are all equally distant from a surface of tranquil 
water is called a line of true level; a straight line tangent to 
the true level at a given point is called the line of apparent 
level with reference to that point. 

Let C be the center of the earth, and A a 
point at its surface ; then AD is the line of true 
level, and AB , the tangent at the point A, is 
the line of apparent level, and BD is the differ¬ 
ence between the true and apparent level for 
the distance AB. 

To compute BD , we will let AC — r, AB C 
= d, and BD = h\ then will BC — r + h. From the right- 
angled triangle BAC, we have (r + h) 2 = r 2 + d 2 , from which 
we get, 

2 rh+h 2 — d 2 . 



But since BD is small, compared with AC, the radius of 
the earth, we may neglect h 2 , whence we shall have, 



2 rh = d 2 , 
d 2 


h = 


2 /•* 


Since 2 r is constant the equation h 


<P 
2 r 


shows that the 








difference between the true and apparent level varies as the 
square of the distance. 

If in the above equation we put for 2r the mean diameter 
of the earth 7,912 miles, and take d = 1 mile, we shall get 
h — 8 inches; hence, at the end of one mile, the true level is 
eight inches below the apparent level. 

If we take any distance in miles, and multiply its square 
by 8, we shall get in inches the difference between true and 
apparent level for distance taken ; thus, for 3 miles we multi¬ 
ply 9 by 8, and get 72 inches, or 6 feet, the correction for 3 
miles. 

To find the correction for 2 chains, or 200 feet, we have, 

Log. 200 = 2.301030 

Log. 5280 = 3.722634 

27578396 

2 


3.156792 

Log. 8 = 0.903090 

— 

Log. .01148 = 27059882 

Which shows that for 200 feet the correction is but little over 
r of an inch. 


To find the distance at which an object , whose height is 
given , can be seen from the surface of the earth. 


Let d and d) be any two distances, and K and h 1 their cor¬ 
responding heights; then we shall have, 


Therefore, 


d l — 2rh 
d 12 = 2 rh 1 . 
d l _ li 

dT 2 ~ h r 


But we know that when d ] — 3 miles, h [ will be 6 feet. 





308 


SURVEYING AND NAVIGATION. 


rpi i< d~ A d A 

Therefore,- = , or — = - 

’9 6 3 2‘ 

Whence, d = L— 

2 

Which gives the distance in miles when the height is given 
in feet. 


Example 1. 

The lantern of the old Eddystone Light-house was 92 feet 
above the water. How far could its light be seen ? 

Put li = 92 feet, and we get, 

d = = 4/138 = 11.75 miles. 

Example 2. 

A spring of water is found to be on an apparent level with 
a given point, and distant from it 15,000 feet. What is the 
fall from the spring to the given point % 

Solution. —Log. 15,000 = 4.176091 

Subtract Log. 5280 to reduce to miles = 3.722634 

0.453457 

Multiply by 2 to square distance 2 

0.906914 

Add log. correction for 1 mile in feet = 1.823909 

Log. 5.38 feet = 0.730823 

which is the fall from the spring to the given point. 

PROBLEM I. 

To find the difference of level between any two stations with 
the theodolite. 

Place the theodolite at one of the stations, say the lower, 
and take the angle of elevation and measure the distance from 









TOPOGRAPHICAL SURVEYING. 


309 

one station to the other; then multiply the measured distance 
by the sine of the angle of elevation ; the product will be the 
elevation of one station above the line of apparent level pass¬ 
ing through the other. 

The same distance multiplied by the cosine of the angle of 
elevation will give the horizontal distance, which reduced to 
miles, squared, and multiplied by 8, will give the correction 
in inches for true level. 

If the horizontal distance is measured, we must multiply by 
the tangent of the angle of elevation for the difference ot 
level. 

Example. 

At a given station an object 6000 feet distant gave with the 
theodolite an angle of elevation of 5° 10'. What was its true 
altitude ? 

Solution. Sin. 5° 10' = 8.954499 

Log. 6000 = 3.778151 

Log. 540.32 = 2.732650 

which gives the elevation of the object above the line of ap- 
parent level, 540.32 feet. 

And cos. 5° 10' == 9.998232 

Log. 6000 = 3.77815 1 

Log. 5975.6 ft. = 3.776383 

To reduce to miles, Log. 5280 = 3.722634 

0.053749 

Square - 2 

0.107498 

Log. T % L823909 

Log. .8539 ft. T.931407 

Whence we may take .85 feet as the correction for true level; 
this added to 540.32 will give 541.17 feet for the altitude of 
the object. 







,-.J~ i* ✓ 

310 SURVEYING AND NAVIGATION. 


PROBLEM II. 


To find the difference of level between any two points on the 
earth?s surface. 



Let A and G be two proposed points, to find the elevation 
of A above G. Measure from A toward G any convenient 
distance, say two chains to C; put a pin at A, and one at C: 
set up the engineer’s level at B , midway between A and C ; on 
the pins at A and C set up the leveling rods; direct the teles¬ 
cope of the level to the target at m, read off from the rod the 
elevation of m above A , and record it in feet and tenths as a 
bach sight ; then direct the telescope to the target at n, read 
off from the rod the elevation of n above c, and record it in 
feet and tenths as a fore sight. From C measure toward G , a dis¬ 
tance CE\ equal to AC ', and put a pin at E, set up the level at 
D, midway between C and E\ and move the rod from A to the 
pin at E\ direct the telescope to the target at P, and read off 
from the rod the elevation of P above (7, and record it as a 
bach sight y then direct the telescope to the target at g , and 
read off the elevation of g above E\ and record it as a fore 
sight. Proceed in this way until it is convenient to set up a 
rod on a pin at G , then find the sum of the back sights and 
and the sum of the fore sights. The difference of these sums 
wilbgive the elevation of one point above the other. 

A horizontal line passing through either of the proposed 
points is called a “datum line,” and any point carefully deter¬ 
mined with reference to other points is called a u bench.” It is 



















I 


TOPOGRAPHICAL SURVEYING. 311 

convenient to record the field notes in the form of a tablet as 
follows: 


STA¬ 

TIONS. 

BACK¬ 

SIGHTS. 

FORE¬ 

SIGHTS. 

DIFFER¬ 

ENCE. 

FROM DATUM 

LINE 

• 

FROM GRADE 

LINE. 

B 

3.2 

6.0 

2.8 

2.8 

1.33 

B 

5.1 

9.0 

3.9 

6.7 

1.56 

F 

4.2 

9.9 

5.7 

12.4 

0.00 


The first column shows the several stations; the second col¬ 
umn shows the back-sights at each station ; the third column 
shows the fore-sights ; the fourth column shows the difference 
of elevation between each two stations ; the fifth column shows 
the distance of each pin from the datum line• the sixth 
column shows the distance of each pin from the grade line. 

If at station B we find Am equal 3.2 feet, and Cn equal 
6.0 feet; and if at station D we find Op equal 5.1 feet, and Eg 
equal 9.0 feet, and at station F if Er equal 4.2 feet, and Gs 
equals 9.9 feet, then will C be 2.8 feet below A , and E will be 
3.9 feet below <7, and G will be 5.7 feet below E. The num¬ 
bers in the fifth column show that C is 2.8 feet below the 
datum line that passes through A, and that E is 6.7 feet below 
the same line, and that G is 12.4 feet below the same line. 

The numbers in the sixth column show that the pin at C is 
1.33 feet above a grade line from A to (7, and that the pin at 
if is 1.56 feet above the same grade. 

The numbers in the fourth column are found by subtracting 
the back sights from the fore sights; the numbers in the fifth 
column are found by adding the numbers in the fourth column. 

To find the numbers in the sixth column, compute the dis¬ 
tance from the datum line to the grade line at each of the 
pins; this can easily be done since we know the position of 
the lines, and the distance from one pin to another. The dif¬ 
ference between these computed distances, and the correspond¬ 
ing numbers in the fifth column, will give the required nuin- 


* 















312 


SURVEYING AND NAVIGATION. 


bers in the sixth column. In the above example, the fore sights 
exceed the back sights ; if at any station the fore sight is less 
than the back sight, the difference in the fourth column must 
be marked, so that in using it we shall get correct numbers 
for the fifth column. 


Example 1. 

From the following field-notes it is required to find the dif¬ 
ference of level, to exhibit the section, and to reduce the same 
to grade. 


STATIONS. 

BACK 

SIGHTS. 

FORE 

SIGHTS. 

* 

DIFF. 

LEVEL. 

FROM 

DATUM. 

FROM 

GRADE. 

1 

8.2 

5.4 

+ 2.8 

+ 2.8 

+ 2.0 

2 

12.1 

9.2 

+ 2.9 

+ 5.7 

+ 4.1 

3 

2.0 

12.1 

-10.1 

-4.4 

-6.8 

4 

7.4 

8.8 

- 1.4 

-5.8 

-9.0 

5 

12.4 

2.6 

+ 9.8 

+ 4.0 

0.0 


Find the numbers in the fourth column by taking the differ¬ 
ence of the fore and back sights, making the remainder posi¬ 
tive when the back sights are the larger, and negative when 
the back sights are the smaller. 

Find the numbers in the fifth column by adding the num¬ 
bers in the fourth column, according to their signs, as follows: 
2.8, the first number in the fourth column, is the first number 
in the fifth column; 5.7, the second number in the fifth col¬ 
umn, is the sum of 2.8 and 2.9; —4.4, the third number, is 
the sum of 5.7 and —10.1; —5.8, the fourth number, is the 
sum of —4.4 and —1.4; and + 4.0, the last number in the 
column, is the sum of —5.8 and -f 9.8. 

To exhibit the section from the above, draw a datum line 
through A, as the first point; on this line, with any scale of 
equal parts, set off the distances of the several pins as at 1, 2, 
3, 4, 5. The numbers in the fifth column of the above tablet 














TOPOGRAPHICAL SURVEYING. 


313 



express the distance of the pins above or below the datum line, 
according as they are positive or negative. Hence, if at the 
points 1, 2, 3, 4, 5, with any convenient scale, we make off¬ 
sets, corresponding to the numbers in the fifth column, a line 
drawn through the extremities of the offsets will exhibit the 
variations in the surface of the ground. 

If a grade line be drawn from A to the pin at 5, it will be 
0.8 above 1 in the datum line, 1.6 above 2, 2.4 above 3, 
and 3.2 above 4, when the pins are equally distant from each 
other. These numbers, combined with the numbers in column 
fifth, give the numbers in column sixth, which show the posi¬ 
tion of the pins with reference to the grade line. 


Example 2. 

FIELD NOTES. 


STATIONS. 

BACK SIGHTS. 

FORE SIGHTS. 

1 

3.35 

2.25 

2 

4.40 

1.80 

3 

2.00 

6.50 

4 

3.25 

4.00 

5 

4.00 

5.00 

6 

5.10 

7.20 


From the above field-notes, required the difference of level 
of the several points, and the cutting necessary to carry a grade 
line from the first to the last pin. 


l 














SURVEYING AND NAVIGATION. 


314 


Example 3. 

FIELD NOTES. 


STATIONS. 

BACK SIGHTS. 

FORE SIGHTS. 

1 

4.32 

7.21 

2 

5.22 

8.17 

3 

9.18 

6.27 

4 

6.27 

6.12 

5 

6.12 

3.76 

6 

9.81 

11.62 

7 

8.47 

9.02 

8 

2.64 

8.91 

9 

1.0T 

7.38 

10 

4.29 

5.32 

11 

5.32 

4.85 

12 

4.85 

3.17 

13 

8.22 

1.53 


From the above field-notes it is required to determine the' 
difference in the elevations of the several points, and also to 
reduce them to a grade line passing from the first to the last 
pin. 


CONTOUR OF GROUND. 

Contour of ground is shown on maps, by marking where 
equi-distant parallel planes meet the surface. We shall give 
only the general principle. 

Let A be the top of a hill, whose contour we wish to delin¬ 
eate; measure any convenient line AB , up or down hill, and 
by the level or theodolite, ascertain the relative elevations of 
a , b, <?, d , &c., as many planes as we wish to represent. 

At a , place the level or theodolite, and level it ready for 
observation ; measure the height of the instrument, and put 
the target on the rod at that height. 














TOPOGRAPHICAL SURVEYING. 


oI5 



C 


Send the rod-man 
and axe-man round 
the hill, on the same 
level as the instru¬ 
ment. Let the rod- 
man set the rod ; the 
Jeveler will sight to it 
through the telescope, and if the target is below the level, he 
will motion the rod-man up the hill, if too high, down the hill ; 
at length he will get the same level, and there the ax-man 
will drive a stake. In the same manner we will establish 
another stake further on ; and thus proceed from point to 
point. To get round the hill, it may be necessary to move 
the instrument several times. The plane thus established, is 
represented by the curve am. 

In the same manner, by placing the instrument at b , we can 
establish the next plane bn. 

Then the next, and the next, as many as we please. Where 
the hill is more steep, two of these parallel planes will bo 
nearer together in the figure; where less steep, they wilt 
appear at a greater distance asunder; and this, with the propel 
shading, will give a true representation of the ground. 

Another method is to select an elevated point in the field 
whose contour is to be represented, and from that point run 
diverging lines ; then with the theodolite or level, determine 
the difference of level between all the important points on 
these lines; then by proportion, ascertain where pins must bo 
driven to mark the intersections of these lines by equi-distant 
parallel planes. Curves drawn through the points marked by 
the pins will indicate the contour of the field. 

Another method is to ascertain the difference of level on 
several parallel sections running through the field. 

The sections plotted will give the contour of the surface 
more or less exactly, according to the number of the sections 
and the nature of the surface. 






31G 


SURVEYING AND NAVIGATION. 




ELEVATIONS DETERMINED BY ATMOSPHERIC PRESSURE, 
AS INDICATED BY THE BAROMETER. 

The higher we ascend above the level of the sea, the less is 
the atmospheric pressure (other circumstances being the same), 
and therefore we can determine the ascent, provided we can 
accurately measure the pressure, and know the law ot its 
decrease. 

The pressure of the atmosphere at any place, is measured 
by the height of a column of mercury it sustains in the bar¬ 
ometer tube. 

It is found by experiment , that air is com¬ 
pressible, and the amount of compression is 
always in proportion to the amount of the 
compressing force. 

Now, suppose the atmosphere to be divided 
into an indefinite number of strata, of the same 
thickness , and so small that the density of each 
stratum may be considered as uniform. 

Commence at an indefinite distance above the 
surface of the earth, as at A, and let w represent 
the weight of the whole column of atmosphere 
resting on A. Let the small and indefinite distances between 
AB, BC , CD , &c., be equal to each other, and we shall call 
them units of some unknown magnitude. 

The weight of the column of atmosphere supposed to rest 
on B , is greater than w, by some indefinite part of w, say the 

Titli part. Then the weight on B , must be expressed by (w-j- v \ 



In the same manner, the weight or pressure resting on C 
must be the weight above B , increased by its nt\\ part ; that 

, il must be {^ — + C — 2 ^ w, which by addition is 

(n+l) 2 w 
•• - — ■ ■■ ■ • 















topographical surveying. 


>17 


In the same manner, we find that the weight resting on D , 

()l 4- l') 3 

must he k- L-w , and so on. For the sake of perspicuity, 


n- 


we recapitulate. 


The 

pressure 

on A is w ; 

units from A 0 

u 

u 

on B is (^—\ w ; 

u 

“ 1 

( 

u 

on G is ( ?? + 1 ) w . 
n* 

u 

“ 2 

6 

u 

on JL) is ' J w ; 

n 3 

u 

“ 3 

i 

a 

~Tjl * (^+1) 4 

on A is v J w ; 

n* 

u 

“ 4 


u 

tt . (n-fl) 5 
on A is v ’ w ; 

nP 

u 

“ 5 


Here we observe the series which represents the pressure of 
the atmosphere, at the different points A, jB, &c., is a series 
in geometrical progression , and it corresponds with another 
series in arithmetical progression / therefore, by the nature of 
logarithms, the numbers in the arithmetical series may be 
taken as the logarithms of the numbers in the geometrical 
series. 

But this system of logarithms may not be hyperbolic nor 
tabular ; indeed it is neither. The base of this system is as yet 
unknown; but our investigations will soon lead to its dis¬ 
covery. 

Now, let the number of units from A to S (the surface of 
the sea), or to the lower of two stations , be represented 
by x ; then the expression for the pressure of the air would be 



But this is neither more nor less than the weight of 


the column of mercury in the barometer, which is sustained 
by this pressure. 

By calling this b , and designating the logarithms of this un¬ 
known system by L' we shall have 


L'b = x. 


a) 








318 


SURVEYING AND NAVIGATION. 


Taking y to represent the number of units from A to T, 
and hi to represent the pressure of the air at that point, we 
shall have 

m,= y • ( 2 ) 

Subtracting (2) from (1), we shall have 

Eb-Z'b, = x-y = sv. 

That is, a certain peculiar logarithm of the barometer col¬ 
umn at the lower station, diminished by the logarithm of the 
barometer column at the upper station will give the difference 
of level of the two stations. But still all is indefinite and un¬ 
known, because we know nothing of these logarithms. 

In algebra, we learn that the logarithms of one system can 
be converted into another by multiplying them by a constant, 
multiplier called the modulus of the system, therefore 

Assume Z to be the modulus or constant that will convert 
common logarithms into these peculiar logarithms. 

Then, ^(log. log. h ( ) = SV. (3) 

Here log. h denotes the common logarithms of the barometer 
column. 

Equation (3) is general, and determines nothing until we 
know S V in some particular case. 

Taking SV some biown elevation , and observing the alti¬ 
tude of the barometer column at both stations, then equation 
(3) will give Z once for all. 

Putting h to represent the biown elevation , we have, in 

_ h_ _ 

log.5—log A, 

Example. —At the bottom and top of a tower, whose height 
was 200 feet, the mercury stood in the barometer as follows: 

At the bottom, 29.96 inches = h 
At the top, 29.7-1 inches = & ( , 

the temperature of the air being 49° of Fahrenheit’s thermo* 
meter. 





TOPOGRAPHICAL SURVEYING. 


319 


Whence, 

200 _ 200 
log. 29.96 — log. 29.74 0.003201 

“But this multiplier is constant only when the mean tem¬ 
perature of the air at the two stations is the same; and for a 
lower temperature the multiplier is less, and for a higher it is 
greater. A correction , however , may be applied for any devia¬ 
tion from an assumed temperature , by increasing or diminish¬ 
ing (according as the temperature is higher or lower) the 
approximate height by its 449 th part for each degree of Fah¬ 
renheit's thermometer. We can moreover change the multi- 
plier to a more convenient one by assuming such a tempera¬ 
ture as shall reduce this number to 60000 instead of 62500. 
Now 62500 exceeds 60000 by its 25th part; and, since 1° 
causes a change of one 449th part, the proportion 


= 62500 nearly 


i • 1° 

44^ * x 


1 _ 

'Z 5 


17.9, 


gives 18° nearly for the reduction to be made in the tempera¬ 
ture of the air at the time of the above observations, in order 
to change the constant multiplier from 62500 to 60000, or to 
10000 , by calling the height fathoms instead of feet. Thus, 
instead of the thermometer standing at 49°, we may suppose 
it to stand at 49° —18° or 31°; and then, we take 10000 as 
the multiplier, and apply a correction additive for the 18° 
excess of temperature.” 

The same observations, for example, being given, to find 
the height of a tower. 

Log. 29.96 = 1.47654 

Log. 29.74 = 1.47334 

DifF. of log. = 0.00320 

Multiplier = 10000 

Product = 32 


Then , the height of the tower is 32 fathoms, or 32 x 6 = 192 
feet, on the supposition that the temperature of the air was 






320 


SURVEYING AND NAVIGATION. 

V / 

31° in place of 49°. But it being 49°, we must increase 192 
by its 4 ^ part for each degree above 31°, that is, by or 35 
nearly of its approximate height, which gives nearly 8 feet to 
add to 192, making 200 feet for the height of the tower. 

The same method is applicable to other cases whatever be 
the temperature of the air at the two stations, provided it be 
die same or nearly the same at both stations, or provided we 
take the mean temperature of the two stations. We can find 
the difference of levels of two stations to considerable accuracy 
by the following 

RUL E. 

1 st. Take the difference of the logarithms of the two baro¬ 
meter columns , and remove the decimal point four places to 
the right . This is the approximate difference of levels in 
fathoms. 

2 d. Add 4 4 tj of the approximate height for each degree of 
temperature above 31°, and subtract the same for each degree 
below 31° ; the result cannot be far from the truth. 

PRACTICAL APPLICATIONS. 

Example 1 . —The barometer at the base of a mountain stood 
at 29.47 inches, and taking it to the top it fell to 28.93 inches. 
The mean temperature of the air was 51°. What was the 
height of the mountain in feet 1 Ans. 503.34 feet. 

Log. 29.47 = 1.469380 
Log. 28.93 = 1.461348 

0.008032. 

Approximate height in fathoms, 80.32 

Correction— Add of the 80.32 to itself, that is, add 3.58 

Height, in fathoms = 83.9 

Multiply by 6 

Height in feet = 503.4 


\ 




X 


TOPOGRAPHICAL SURVEYING. 321 

The average height of the barometer, at the level at the sea, 
is 30.09 inches; and now, if we know the average height of 
the barometer at any other place, and the average tempera¬ 
ture, it is equivalent to knowing the elevation of the latter 
place above the level of the sea. 

For example, the mean height of the barometer at Albany 
Academy is 29.96, and the mean temperature is 49°. How 
high is the academy above the tide water? 

A ns. 117.3 feet. 

Log. 30.09 = 1.478422 49° 

Log. 29.96 = 1.476542 31 

0.001880 18° 

Approximate height = 18.80 fathoms. 

Add iW or j' s , _75 

19.55x6 =r 117.3 feet 

When the difference of temperatures at the two stations is 
considerable, the result must be affected by it. 

When the upper station is the coldest, which is generally the 
case, the mercurial column will be shorter than it otherwise 
would be, and consequently indicate too great a height. 

If the temperature of the upper station is taken for the tem¬ 
perature of the lower, the mercurial column at the lower sta¬ 
tion would not be high enough, and the deduced result would 
be too small, as is the case in example 5. 

The contraction of mercury being about one 10000th part 
for each degree of cold, or 0.0025 in a column of 25 inches, it 
would require 4° difference of temperature to produce an effect 
amounting: to one division on the scale of a common barometer, 
where the graduation is to hundredths of an inch. 

This correction is combined with the former in the follow- 
ing equation, in which t and t\ represent the temperatures of the 
air at the two stations (t at the lower station), q and q 1 the 
temperatures of the mercury at the two stations, as indicated 
by the attached thermometer 





322 


SURVEYING AND NAVIGATION. 


The fraction 0.00223 is equal to nearly; h = the height 
sought, b and b t represent the observed heights of the mercurial 
columns. 


h = 10000 


1 + 0.00223 



j _ b_ _ 

n b(l + . 0001 )(^— q') 


Beside the corrections previously considered, regard is some¬ 
times had to the effect of the variation of gravity in different 
latitudes, and at different elevations above the earth’s surface. 
The latter, however, is too small to require any notice in an 
elementary work. The former may be found by multiplying 
the approximate height by 0.0028371 xcos. 2 lat. It is addi¬ 
tive, when the latitude it less than 45°, and subtractive when 
greater. Or it may be taken from the following table. 


Latitude. 

Correction. 

0 °. 

.+ "532 °I 

5°. 


10 °. 


15°. 

.+ 4 <5 ? 

20 °. 

.+ 460 

25°. 


30°. 


35°. 


40°. 


45°. 

. 0 

50°. 

I 

55°. 

_ 1 

60°. 

1 

65°. 


70°. 


75°. 

_ 1 

80°. 

_ 1 

85°. 

I 

90° . 



of the app. height. 























TOPOGRAPHICAL SURVEYING. 


323 


Example 2.—Given, the pressure of the atmosphere at the 
bottom of a mountain, equal to 29.68 in. of mercurj, and that 
at its summit, equal to 25.28 in., the mean temperature being 
50°, to find the elevation. 

Ans. 727.2 fathoms, or 4363.2 feet. 

Example 3.—The following observations were taken at the 
foot and summit of a mountain, namely, 

At the foot, 

Bar. 29.862 ; attach, therm. 78° ; detach, therm. 71°. 

A.t the summit, 

Bar. 26.137; attach, therm. 63°; detach, therm. 55°. 

It ;s required to find the elevation. 

A ns. 612.9 fathoms, or 3677.4 feet. 

Example 4.—It is required to find the height of a mountain 
in latitude 30°, the observations with the barometer and ther¬ 
mometer being as follows: namely, 

At the foot, 

Bar. 29.40; attach, therm. 50°; detach, therm.* 43°. 

At the summit, 

Bar. 25.19; attach, therm. 46°; detach, therm. 39°. 

Ans. 683.27 fathoms, or 4097.62 feet. 

Note. —If we assume any temperature, for instance 45 3 , and the height 
of the barometer at the level of the sea, at 30.09 inches, we can compute 
the elevation of the point, where it would be 29.99, 29.89, 29.79, 29.69, &c\, 
inches; and thus we might form a table, showing the elevations that would 
correspond to any assumed height of the barometer at that temperature. 
It will be found, that the first fall of T \ of an inch will correspond to about 
88 feet in elevation; but every subsequent tenth will require a greater i n 1 
greater elevation. 

Example 5.—At a certain station the average reading of 
the barometer, reduced to a temperature of 32°, was found to 

* The attached thermometer measures the temperature of the mercury 
in the barometer, and the detached thermometer, that of the surrounding 


air. 


324 


SURVEYING AND NAVIGATION. 

be 28.130 inches. What is the elevation of the station above 
tide water? Ans . 1755 feet. 

Example 6.—Where the average reading of the barometer, 
reduced to 32°, is 28.980 inches, what is the elevation above 
the sea? Ans . 979 feet. 

Example 7.—On a mountain the barometer indicated 24.860 
inches, and the thermometer stood at 68°. What was the 
elevation of the mountain. Ans . 5386 feet. 












/ 


CHAPTER VI. 


NAVIGATION. 


SECTION I. 

DEFINITIONS. 

Navigation is the art of determining the position of a ship 
on the ocean. 

The Axis of the earth is the line about which it revolves ; 
the extremities of the axis are called poles. 

The Equator is a circle whose plane is perpendicular to the 
axis of the earth. 

Meridians are great circles passing through the poles of the 
equator. 

Longitude is measured on the equator from an assumed 
meridian. 

Latitude is measured on a meridian north or south of the 
equator. 

Parallels of Latitude are small circles parallel to the 
equator. 

Difference of Latitude is the arc of a meridian between two 
parallels of latitude. 

Departure is distance measured on a parallel of latitude, 
east or west from a given meridian. 



326 


SURVEYING AND NAVIGATION. 


In Navigation the surface of the ocean is assumed to be 
spherical. 

The Course of a ship is the angle its path makes with a 
meridian. 

A ship is said to sail on the same course when its path 
crosses the meridians at the same angle. 

The Rhumb Line is the path of a ship pursuing a uniform 
course on the surface of the ocean. 

Nautical Distance is the portion of the Rhumb line inter¬ 
cepted between two points. 

When a ship sails on several courses, the course and dis¬ 
tance from the point left to the point arrived at are called the 
course and distance made good. 

The several courses and distances spoken of together are 
called a traverse / and computing the course and distance made 
good, is called working a traverse. 

The Horizon of a place is a circle whose plane touches the 
earth at the place. 

T1 le Zenith is the upper pole of the horizon. 

Vertical Circles are those circles that pass through the poles 
of the horizon. 

The Altitude of a body is the arc of a vertical circle inter¬ 
cepted between the horizon and the body. 

The Ecliptic is the great circle in whose plane the earth 
performs its revolution around the sun. 

T1 le Equinoctial Points are the intersections of the ecliptic 
and equator. 

One is called the vernal , the other the autumnal equinox. 

Right Ascension is measured on the equator eastward from 
the vernal equinox. 

Declination is measured on a meridian north or south of the 
equator. , . 



NAVIGATION. 


327 


Tlie Meridian of any place is the great circle that passes 
through the poles of the equator, and the poles of the horizon 
of the place. 

The course of a ship is ascertained by the Mariner’s Com¬ 
pass, which differs from the surveyor’s compass only in its 
graduation and adjustments. 


LEEWAY. 

The angle included between the direction of the fore and 
aft line of a ship, and that in which she moves through the 
water, is called the leeway . 

When the wind is on the right hand side of a ship, she is 
said to be on the starboard tack ; and when on the left hand 
side, she is said to be on the larboard tack ; and when she 
sails as near the wind as she will lie, she is said to be close 
hauled. Few large vessels will lie within less than six points 
to the wind, though small ones will sometimes lie within about 
five points, or even less; but, under such circumstances, the 
real course of a ship is seldom precisely in the direction of her 
head; for a considerable portion of the force of the wind is 
then exerted in driving her to leeward, and hence her course 
through the water is in general found to be leeward of that 
on which she is steered by the compass. Therefore to deter¬ 
mine the point toward which the ship is actually moving, the 
leeway must be allowed from the wind, or toward the right 
of her apparent course when she is on the larboard tack; but 
toward the left , when she is on the starboard tack. 

It is only when a ship crowds to the wind, that leeway 
is made. 

It is seldom that tw r o ships on the same course make pre¬ 
cisely the same leeway; and it not unfrequently happens, 
that the same ship makes a different leeway on each tack. 
It is the duty of the officer of the watch to exercise his 
best skill in determining or estimating how much this devia¬ 
tion from the apparent course amounts to. 


328 


SURVEYING AND NAVIGATION. 


CARD OF THE MARINER’S COMPASS. 



The card attached to the needle of the mariner’s compass is 
divided into thirty-two equal parts, called points. There are 
eight points in each quadrant; they are counted from the 
meridian NS, as in the diagram; thus N.b.E. is one point east 
of north, and is read north by east; the angle is one-eighth of 
90°, or 11° 15 f . And N.N.E. is two points east of north, or 22° 
30' east, and is read north, north east. In the same manner 
the courses in the other quadrants are read. Each point is 
usually subdivided into quarter points of 2° 4S f 45" each. Sea¬ 
men take the course of their ship to the nearest quarter point, 
and for their convenience in practical navigation, the traverse 
table is arranged so as to give for various distances the lati¬ 
tudes and departures for points and quarter poiute. and on the 
plane scale the line of chords, called the lint} of rhumbs, is 
marked for the same angles. 







NAVIGATION. 


320 


TA B L E. 


Showing the degrees and minutes corresponding to each 
point and quarter point of the compass. 


N. 

S. 

S. 

N. 

POINTS. 

ANGLES 

NAE. 

S.|E. 

SAW. 

N IW. 

4 

2° 48' 45" 

N.iE. 

SAE. 

SAW. 

NAW 

1 

• o 

5 ° 37' 30" 

N.fE. 

a?E. 

s.Jw. 

N.fW. 

3 

•¥ 

8° 20' 15" 

N.bE. 

SbE. 

S.bVV. 

N.bW. 

1 

ir 15 ' o" 

N.bE.jE. 

S.bE.’E. 

S.bWdW. 

N.bW.jW. 

lj 

14° 3'45" 

N.bEAE. 

S.bE.iE. 

S.bVV.iW. 

N.bW.iW. 

ii 

16° 52' 30" 

N.bE.$E. 

S.bE. E. 

S.bW. 3 W. 

NbW.JW. 

i, 3 

19 41/ 15" 

N.N.E 

S.S.E. 

s.s.w. 

N.N.W. 

2 

22° 30' 0" 

N.N.E. ^E. 

S.S.E.’E. 

S.S.W.JW. 

NN.W.jW. 

21 

25° 18' 45" 

N.N.E.iE. 

S.S.EAE. 

S.S W.IW. 

N.N.W.iW. 

21 

28° 7'30" 

N.N.E. r, E. 

S.S.E JE. 

S.S.W. 3 W. 

N.N.W.fW. 

2j 

30° 50/ 15" 

N.E.bN. 

S.E.bS. 

S.W.bS. 

N.W.bN. 

3 

33° 45' 0 " 

N.E.9N. 

S.E.JS. 

S.W. 3 S. 

N.W. 3 N. 

3* 

36° 33' 45" 

N.E.iN. 

S.E. 5 S. 

S.W.-JS. 

N.WAN. 


39° 22' 30" 

N.E.jN. 

S.E.jS. 

S.W.]S. 

N.W.iN. 

3j 

42° 11' 15" 

N.E. 

S.E. 

s.w. 

N.W. 

4 

45° 0' 0" 

N.E.jE. 

S.E.fE. 

S.W.’W. 

N.W.iW- 

41 

47° 48' 45" 

N.EAE. 

S.EAE. 

S.W.1W. 

N.WAW. 

41 

50" 37' 30" 

N.E.JE. 

S.E.|E. 

s.w.fw. 

N.W.fW. 

4? 

53 s 26' 15" 

N.E.bE. 

S.E.bE. 

S.W.bW. 

N.WbW. 

5 

56° 15' 0" 

N.E.bE. E. 

S.E.bE.jE, 

S.W.bW.|W. 

N.W.bW.’W. 

5]- 

59" 3'45" 

N.E.bE. iEJ S.E.bE A E. 

S.W.bW.'W. 

N.W.bW.'W. 

5i 

61° 52' 30" 

N.E.bE. § E. 

S.E.bE.jE. 

S. W.bW.fW.:N.W.bW. 3 W. 

5 3 

64° 41' 15" 

E.N.E E.S.E. 

W.S.W. 

W.N.W. 

6 

67 c 30' 0" 

E.bN?N. iE.bSfS. 

WbS. 3 S. 

W.bN. 3 N. 

61 

70° 18' 45" 

E.bN.iN. 

E.bS.-’S. 

W.bS.iS. 

W.bNAN. 

6 2 

73° 7'30" 

E.bN.fN. 

E.bSjS. 

W.bS.fS. 

W.bN.fN. 

6f 

75° 56' 15" 

E.bN. 

E.bS. 

W.bS. 

W.bN. 

7 

78° 45' 0" 

E.?N. 

E.?S. 

W. 3 S. 

W.fN. 

71 

81° 33' 45" 

E.iN. 

E.JS. 

WAS. 

WIN. 

71 

84° 22' 30" 

E.jN. 

E.JS. 

W.jS. 

W.JN. 

n 

87° 11' 15" 


SECTION II. 

PLANE SAILING. 

Plane Sailing' is a method of determining the position of a 
a ship at sea, from the properties of plane triangles. 

Since the path of the ship crosses the meridians at the same 
angle, the distance , departure , and difference of latitude have 













330 


SURVEYING AND NAVIGATION. 


the same relation to each other as the sides of a plane right- 
angled triangle, in which the course is the angle opposite the 
departure and the distance is the hypotenuse. 

In the triangle ABC, if AC represent the 
distance, and the angle at A is the course, then 
will AB represent difference of latitude, and 
BC will represent departure ; any two of these 
quantities being given, we can determine the 
others by trigonometry. (See Chap. 2 , Sect. II., 

Prop. 3). Thus we have 

It \ AC w sin.H : BC\ 
or It : distance : : sin. course : departure ; (1) 

It : AC w cos.H : AB, 

or It : distance : : cos. course : diff. lat.; ( 2 ) 

It : tan.H :: AB : BC, 

or It : tan. course : : diff. lat. : departure. (3) 

EXAMPLES. 

1 . If a ship sail from Cape St. Vincent in lat. 37° 2 ' 54" 
N., 148 miles on a course S. 39° 22 P W, required her lati¬ 
tude and departure. 

By proportion ( 1 ), we have 

Sin. 39° 22 £, = 9.802359 

Log. 148 = 2,17026 2 

Log. 93.89 = log. depart. = 1.972621 

By proportion ( 2 ), we have 

Cos. 39° 22 ,}' = 9.888185 
Log. 148 = 2.1702 62 

Log. 114.4 = 2.058447 

Whence the difference of latitude is 114.4 miles, which is 
1 ° 54' 24". Hence 

Latitude left = 37° 2 f 54 " N. 

1 ° 54' 24" S.* 

Latitude arrived at = 35 ° gf 30 " N 








NAVIGATION. 


33 J 


2 . A ship sails from latitude 40° 28' 14., E. S. E. 21 miles. 
Required her latitude and departure. 

Ans. Latitude 40° 20' 14.; Departure 19.4 miles. 

3. A ship sails from Oporto, in lat. 41° 9' 14., 14.47° 48J'W. 
315 miles. Required her departure and latitude. 

Ans. Dep. 233.4 miles; Lat. 44° 41 f 14. 

4. A ship sails from lat. 55° 1' 14., S. 33° 45' E., till her 

departure is 45 miles. Required her latitude, and the distance 
sailed. Ans. Dist. 81 miles ; Lat. 53° 54' 14. 

5. A ship from lat. 36° 12' 14. sails south-westward, until 
she arrives at lat. 35° 1' 14., having made 76 miles departure. 
Required her course and distance. 

Ans. Course S. 46° 57' W. ; Dist. 104 miles. 

• 6. A ship sails from Halifax, in lat. 44° 44' 14., S. 50° 37P 
E., until her departure is 128 miles. Required her latitude, 
and the distance sailed. 

Ans. Lat. 42° 59'; Dist. 165.6 miles. 

7. A ship leaving Charleston light, in latitude 32° 43' 30" 14., 
sails north eastward 128 miles, and is then found 39 miles 
north of the light. Required her course, latitude, and depar¬ 
ture. Ans. Latitude 33° 22' 30" N .; Course 14. 72° 16' E. ; 

Departure 122 miles. 

8 . A ship from Cape St. Roque, in latitude 5° 10' south, 
sails 14. E. } N. 7 miles an hour, from 3 P. M. until 10 A. M. 
Required her distance, departure, and latitude. 

Ans. Dist. 133 miles; Dep. 84.4 miles; Lat. 3° 27' S. 

7. A ship from latitude 41° 2' 14. sails 14. 14. W. f W. 5j 
miles an hour, for 2^ days; required her distance, departure, 
and latitude arrived at. 

Ans. Dist. 330 miles ; Dep. 169.7 miles ; Lat. 45° 45' 14. 

When a ship has sailed on several different courses, the 
reduction of them to a single course and distance is called 


332 


SURVEYING AND NAVIGATION. 


working the traverse. This is most readily done by taking 
from the traverse table the latitude and departure of each 
distance, and arranging the numbers in columns in a tablet 
ruled for the purpose. Then the difference between the sum 
of the northings and the sum of the southings, will be the lati¬ 
tude of the distance made good ; and the difference between 
sum of the eastings and sum of the westings, will be the 
departure of the distance made good. 

A ship makes the following courses and distances. 



BEARINGS. 

MILES. 

1 

S.bW. 

23 

2 

w.s.w. 

40 

3 

S.lV.fW. 

18 

4 

W.’N. 

28 

5 

S.bE. 

12 

6 

S.S.E.|E. 

16 


Required her course and distance made good, her departure 
and difference of latitude. 


From the traverse table, we obtain the numbers in the fob 
lowing tablet. 



COURSES. 

POINTS. 

MILES. 

• 

N. 

s. 

E. 

w. 

l 

S.bW. 

S. 11° 15' W. 

1 

23 


22.56 


449 

2 

W.S.W. 

S. 67° 30' W. 

6 

40 


15.31 


36 96 

3 

S.W.fW. 

S. 53° 26p W. 
N. 84° 22f W. 

S. 11° 15' E. 

4£ 

18 


10.72 


14.46 

4 

W.^N. 

n 

28 

2.74 



27.87 

5 

S.bE. 

i 

12 


11.77 

2.34 


6 

S.S.E.VE. 

S. 25° 18£' E. 

n 

16 


14.46 

6.84 







2.74 

74.82 

9.18 

83.78 







2.74 


9.18 


12.08 14.60 


"Whence we have the departure 74.60 miles, and the differ¬ 
ence of latitude 72.08 miles. 






































NAVIGATION. 


333 


To find the course, 

Log. 74.60 = 1.872739 
Log. 72.08 = 1.857815 

Tan. 45° 59' = 10.014924 
Whence the course is S. 45° 59' W. 

To find the distance, 

Log. 74.CO = 1.872739 
Sin. 45° 59' = 9.856812 

Log. 103.7 = 2.015927 

Whence the distance is 103.7 miles. 


10 . A ship makes the following courses and distances: 



BEARINGS. 

MILES. 

1 

s.s.w. 

18 

2 

s.w. 

15 

3 

S.W.bS. 

20 

4 

w. 

9 

5 

w.s.w. 

14 


.Required her course and distance made good, the departure 
and difference of latitude. 



COUR8E8. 

POINT8. 

MILKS. 

N. 

8. 

B. 

W. 

l 

S.S.W. 

S. 22° 30' W. 

2 

18 


16.63 


1 

6.89 

2 

s.w. 

S. 45° W. 

4 

15 


10.61 


10.61 

3 

S.W.bS. 

S. 33° 45' W. 

3 

20 


16.63 


11.11 

4 

w. 

West. 


9 


0.0 


9.00 

5 

W.S.W. 

S. 67° 30' W. 

6 

14 


5.36 


12.93 



49.23 


50.54 | 


Whence we see that the departure is 50.54, and the differ¬ 
ence of latitude is 49.23. 









































334 SURVEYING AND NAVIGATION. 


To find the course, 

Log. 50.54 = 1.703635 

Log. 49.23 = 1.692230 

Tan. 45° 45 f = tan. course, 10.011405 

To find the distance, 

Log. 50.54 = 1.703635 

Sin. 45° 45' = 9.855096 

Log. 70.56 = log. distance, 1.848539 
11 . A ship makes the following courses and distances. 



BEARINGS. 

MILES. 

1 

s.s.w. 

42 

2 

s.w. 

18 

3 

w.s.w. 

24 

4 

w. 

11 

5 

KW. 

108 


Required her course and distance made good, her departure 
and difference of latitude. 

Ans. Course, N. 83° 32' W.; Distance, 139.3 miles f Depar¬ 
ture, 138.4 miles; Diff. lat., 15.7 miles. 

12 . A ship on the equator sails as follows: 



BEARINGS. 

MILES. 

1 

E.bS. 

90 

2 

E.^S. 

76 

3 

E.N.E. 

41 

4 

KE. 

. 

82 


Required her position. 

Ans. Course, N. 79° 23' E.; Distance, 264.3 miles; Depar¬ 
ture, 259.8 miles; and the ship is 48.7 miles north of the equator 























NAVIGATION. 


335 



13. A sliip makes the following courses and distances : 



BEARINGS. 

MILES. 

1 

w. 

28 

2 

S.W.bW. 

30 

3 

S.W.bS. 

46 

4 

E.S.E. 

28 


.Required her course and distance made good. 

Ans. Course, S. 38° 43' W .; Distance, 84.1 miles. 

14. A ship from latitude 42° north makes the following 
courses and distances, 



BEARINGS. 

MILES. 

1 

s.s.w. 

48 

2 

S.bE. 

34 

3 

S.W.iW. 

26 

4 

E. 

17 


Required the latitude arrived at, and the course and distance 
made good. 

Ans. Lat. 40° 25' N.; Course S. 8° 22* W.; Dist. 9G.2 miles. 

When in course of the day a ship has sailed on several dif¬ 
ferent tacks, seamen often determine approximately the course 
and distance made good by drawing a diagram with a scale 
and dividers, as in the following example: 

15. A ship makes the following courses and distances. 


1 

S.bE. 

20 

miles 

2 

S.E. 

30 

a 

3 

E.bS.lS. 

42 

a 

4 

N.E. 

36 

a 


Required the course and distance made good, by construction. 




















336 


SURVEYING AND NAVIGATION. 


N 



About A as a center, describe a circle with a radius equal to 
the chord of 60°, taken from the line of chords on the plane 
scale. Draw JVAS as a meridian ; then from S set off $1, 
with the chord of the first course, taken from the line of 
rhumbs; then set off $4 with the chord of the second course, 
taken from the same line; then set off with the chord of 
the third course; then from AT set off A 4 in the northeast 
quadrant with the chord of the fourth course taken from the 
same line of rhumbs ; then draw the lines A 1, A 4, A6±, and 
A4. On Al set oft’the first distance 20 miles, taken from any 
scale of equal parts; suppose it extends to B. Then through B 
draw a line parallel to Ad, and on it set off BC equal to 30 
miles, the second distance; through C draw aline parallel to 
A6J ; on it set off CD equal to 42 miles, the third distance; 
through D draw a line parallel to A 4 in the northeast quad¬ 
rant, and from the same scale of equal parts set off DE equal 
to 36 miles, the fourth distance. Then draw the line AE\ and 
it will represent the distance made good ; applying it to the 
scale, we find for the above example it equals 95 miles. For 
the course, extend the dividers from S to where the line AE 
cuts the circle, and then apply the dividers to the line of 
chords, and we shall get the bearing of AE equal to S. 73° 
10 ' E., nearly. 


i 










NAVIGATION. 


337 


In plane sailing it is not assumed tliat tlie surface of tlie 
ocean is a plane, or nearly so, for sliort distances. But 
tlie properties of tlie rhumb line are such, that crossing the 
meridians each at the same angle, the length of the rhumb 
line , the difference of latitude , and tlie departure , have the 
same relation to each other as the sides of a plane triangle. 

The rhumb line is a spiral; the difference of latitude is an 
arc of a great circle ; the departure is an arc of a parallel of 
latitude, a small circle. It is assumed that these lines are 
drawn upon the surface of a sphere, and in consequence ot 
the above relation, we get the difference of latitude and de¬ 
parture accurately for a single course, whether the distance is 
lon£ or short. 

When a ship sails on several courses the sum of the depar¬ 
tures is not precisely equal to the departure for the same dis¬ 
tance on a single course. 

Thus, when we work a traverse we get the difference of lati¬ 
tude accurately, and the departure approximately. Plane 
sailing does not give difference of longitude. 


SECTION III. 

TO FIND THE DIFFERENCE OF LONGITUDE. 


Let C be the center of the earth, Ptlie 
pole, P(7the axis of the earth, PDA 
and PEB meridians, AB a portion 
of the equator, DE a corresponding 
portion of a parallel of latitude. Then 
will AD be latitude; and if DII is 
parallel to AC\ it will be the cosine of 
the latitude to a radius AC\ DE will 
be departure, and AB will be differ- 










/ 


338 SURVEYING AND NAVIGATION 

ence of longitude. But we have from similar sectors, DEB. 
and ABC\ 

DH\ AC:: BE: AB or, 

(1) Cos. lat. : B :: departure : diff. longitude. 

This proportion can he represented by a triangle as BCD % 
where BC is departure. 

The angle BCD is equal to the latitude, and 
the hypotenuse DC is the difference of longi¬ 
tude. 

From plane sailing, we have 

0 

Sin. course : R :: departure : distance. 

This combined with the preceding proportion so as to eliminate 
departure, will give 

(2) Cos. lat. : distance : : sin. course : diff. longitude. 



Also, from plane sailing, we have, 

Diff. latitude : departure : : R : tan. course. 


And this when combined with 
proportion (1) so as to eliminate 
departure will give, 

(3) Cos. lat. : diff. lat. : : tan. 
course : diff. longitude. 

If we put the triangle for plane 
sailing with the triangle for longi¬ 
tude, we shall have the diagram 
in the margin, in which A6 7 is the 
distance, the angle at A is the 
course, AB is difference of lati¬ 
tude, BC is departure, the angle 
BCD is the latitude, the angle at 
D is co-latitude, and DC is the 
difference of longitude. 







NAVIGATION. 


339 


The triangle DAC gives the second proportion, since we 
have, 

Sin .1) : AC:: sin .A : DC , 

which is equivalent to the second proportion. The third is 
derived by using AD and tan.Al as equivalent to DC. 

Whence we have three propositions for finding longitude. 

1. Cosine of latitude is to radius as departure is to differ¬ 
ence of longitude. 

2 . Cosine of latitude is to distance as the sine of the course 
is to difference of longitude. 

3. Cosine of latitude is to difference of latitude as the tan¬ 
gent of the course is to the difference of longitude. 

The latitude in these proportions for longitude is the arith¬ 
metical mean of the latitude left and arrived at; it is obtained 
by taking half the sum when both latitudes are north or south, 
and half the difference when one is north and the other south. 
It is assumed that, measured on the parallel of this mean lati¬ 
tude, the departure is equal to the meridian distance; this is 
nearly so when the difference of latitude is small, and it may 
be rendered quite exact by corrections taken from a table pre¬ 
pared for the purpose, and first published by Mr. Workman 
in 1805. 

When a ship sails on the parallel of latitude, the distance 
equals the departure, and proportion (1) gives the difference 
of longitude ; this is called parallel sailing. When the ship 
does not sail on a parallel, but changes its latitude, and the 
difference of longitude is determined by proportions (2) and 
(3), the method is called middle latitude sailing. 

EXAMPLE UNDER PROPORTION I 

1 . What difference of longitude corresponds to If mile* 
departure in the latitude of 37° 23' ? Ans. 59.15 miles. 

Let x = the difference of longitude required. 


340 


SURVEYING AND NAVIGATION. 


Then, 


cos. 37° 23' : R. : : 47 : x = 


47 R 


cos. 37° 23' 


Log. 47 E = 11.672098 
Cos. 37° 23' = 9.900144 

Log. Diff. Ion. 59.15 = 1.771954 

2. How many miles, or how much departure, corresponds to 
a degree in longitude on the parallel of 42° of latitude ? 

Ans. 44.59 miles. 


Here the longitude of one degree is given. 

60 cos. 42° 


E : cos. 42° : : 60 : x = 


E 


Log. 60 = 1.778151 
Cos. 42° = 9.871073 

Log. 44.59 = 1.649224 

i 

3. A ship sails east from Cape Race, 212 miles; required 
her longitude. The latitude of-the cape is 46° 40' X., longi¬ 
tude 53° 3' 15" west. Arts. Lon. 47° 54' west. 


4. Two places in lat. 50° 12 f differ in longitude 34° 4S'; 
required their distance asunder in miles. Ans. 1336. 


5. How far must a ship sail W. from the Cape of Good Hope 
that her course to Jamestown, St. Helena, may be due north? 

Ans. 1193 miles. 


Note 


.—Cape | 


lat. 34° 29' S. 
Ion. 18° 23' E. 


Jamestown 


t lat. 15° 55' S. 
i Ion. 5° 43' 30" W. 


6. How far must a ship sail E. from Cape Horn to reach 

the meridian of the Cape of Good Hope ? The latitude of 
Cape- Horn being 55° 58' 30" S., Ion. 67° 21' W., and the lati¬ 
tude and longitude of the Cape of Good Hope being as stated 
in the above note., Ans. 2878 miles. 

7. In what latitude will the difference of longitude be three 

times its corresponding departure? Ans. 70° 31 f 44'. 






NAVIGATION. 


341 


The following examples are designed to illustrate the prin¬ 
ciples of Section 3. 


Example 1. 

A ship from Cape Clear, Ireland, in latitude 51° 25* JI. and 
longitude 9° 29' W., sails as follows: 



BEARINGS. 

MILES. 

1 

S.S.E.JE. 

16 

2 

E.S.E. 

23 

3 

S.W.bW.^V. 

36 

4 

WfK 

12 

5 

S.E.bE.jE. 

41 


Required her course, distance, difference of latitude, and 
difference of longitude. 


TRAVERSE TABLE. 












DIFF. 

LAT. 

DEPARTURE. 

COURSES. 

POINTS. 

DIST. 





N. 

s. 

E. 

w. 




S.S.E.JE. 

21 

16 


14.5 

6.8 


E.S.E. 

6 

23 


8.8 

21.2 


S.W.bW.*W. 


36 


17.0 


31.7 

W.fK 

7| 

12 

1.8 



11.9 

S.E.bE.^E. 

51 

41 


21.1 

35.2 


# 



1.8 

61.4 

63.2 

43.6 





1.8 

43.6 



Result, 59.6 19.6 


Lat. left = 51° 25' N. 
Diff. lat. = 1° 00' S. 


Lat. in = 50° 25' K 


Mid. lat. 50° 55'. 








































342 SURVEYING AND NAVIGATION. 
To find the course and distance, by trigonometry, 


As diff. lat., 59.6 miles, 

1.775246 

: Radius 90° 

10.000000 

: : Dep.,19.6 miles, 

1.292256 

: Tan. course, 1S° 12', 

9.517010 

As sin. course 18° 12' 

9.494621 

: Dep.,19.6 miles, 

1.292256 

: : Radius 

10.000000 

: Dist. 62.75 miles, 

1.797635 

To find the difference of longitude. 

As cos. 50° 55' 

9.799651 

: Radius 

10.000000 

: : Dep.,19.6, 

1.292256 

: Diff*. Ion., 31.09 miles, 

1.492605 


Longitude left = 9° 29 f west 
Diff. Ion. = 31* east 

Lon. in = 8° 58' west. 

Thus, we have found the course 18° 12 f ; distance, 62.75 
miles ; dift*. longitude, 31' E.; lat. in, 50.25 N.; Ion., 8° 58' W. 

If these be the distances run in a day, from noon to noon 
again, then the preceding operation is called working a day’s 
work ; otherwise it is called working a traverse. 

Example 2. 

A ship sails from lat. 37° 2' north, and longitude 9° 2 f west, 
and makes the following courses and distance. * 



BEARINGS. 

MILES. 

1 

E.S.E. 

45 

2 

S.W.bW. 

43 

3 

S.E.bS. 

64 

4 

N.N.E. 

22 















NAVIGATION. 


343 


Required the latitude and longitude arrived at, and the course 
and distance made good. 

Take the northings, southings, eastings, and westings, from 
the traverse table, and arrange them as in the following tablet: 



COURSKS. 

POINTS. 

H 

in 

5 

N. 

s. 

E. 

w. 

1 

E.S.E. 

6 

45 


17.2 

41.6 


2 

S.W.bW. 

5 

43 


23.9 


35.8 

3 

S.E.bS. 

3 

64 


53.2 

35.6 


4 

N.N.E. 

2 

22 

20.3 


8.4 






20.3 

94.3 

85.6 

35.8 






20.3 

35.8 



74.0 49.8 

Whence the difference of latitude is 74 miles south, 
which is 1° 14 f . 

And the given latitude is 37° 2 f X. 

Difference of latitude, 1° 14 f S. 

Latitude arrived at, 35° 48' N. 

The departure is 49.8 miles; the mean latitude is 36° 25 f , 
half the sum of the two above given ; hence by Proportion 
1, we have 

Cos. 36° 25 f : Jl :: 49.8 : diff. Ion. 

Therefore, Log. 49.8 = 1.697229 

Cos. 36° 25' = 9.905645 

Log. 61.88 1.791584 

And we have 61.88, or nearly 62, miles for the difference of 
longitude, which is equal to 1° 2'. 

Therefore, longitude given = 9° 2 f W. 

Difference of longitude = 1° 2 r E. 

Longitude arrived at 


= 8 ° 0 ' W. 



































344 SURVEYING AND NAVIGATION. 

To find the course,, we Lave 

Log. 49.8 = 1.697229 

Log. 74.0 = 1.869232 

Tan. 33° 56' 9^827997 

Whence the course is S. 33° 56' E. 

To find the distance,, we Lave 

Log. 49.8 = 1.697229 

Sin. 33° 56' = 9.74 6812 
Log. 89.2 1.950417 

Whence the distance is 89.2 miles. 

Example 3. 

A ship in latitude 33° 56 r south, and longitude 18° 23* east, 
sails 

1. N.W.bN. 12 miles. 

2. N.W. 36 “ 

3. N.W.bW. 140 u 

Required her latitude and longitude, and the course and 
distance made good. 

Ans. Lat. 32° 3' S.; Lon. 15° 26' E.; Course, K. 52° 41 f W.; 
Dist. 187 miles. 

Example 4. 

A ship in latitude 42° 40' N., longitude 59° W., sails S.E.bS, 
600 miles. Required the latitude and longitude arrived at. 

Ans . Lat. 34° 21 f N.; Lon. 51° 53' W. 

Example 5. 

A ship in latitude 51° 18' N., longitude 11° 15' W., sailed 
S.E.BS. 480 miles. Required the latitude and longitude 
arrived at. Ans. Lat. 45° 22' N.; Lon. 3° 10' W. 





NAVIGATION. 


345 


Example 6 . 

A ship finds by observations upon the light at Sandy Hook 
that her latitude is 40° 25' N., and her longitude is 74° W. 
She then sails S.bAY. 520 miles; it is required to find her 
course and distance thence to Nassau in latitude 25° 4' N., 
and longitude 77° 18' AY. 

Ans. Course, S. 8° 45' AY.; Dist. 416 miles. 

Example 7 . 

Required the course and distance from Land's End, in lati¬ 
tude 50° 6' N., and longitude 6° AY., to Bermuda, in latitude 
31° 20' N., longitude 64° 48' AY. 

Ans. Course, S. 66° 56' AY.; Hist. 2874 miles. 

Example 8 . 

Required the course and distance from latitude 37° 48 N., 
and longitude 25° 13' AY., to latitude 50° 13' N. and longitude 
3° 38' AY. 

Ans. Course, N. 51° 11' E.; Distance, 1189 miles. 

Example 9 . 

A ship in latitude 37° 48' N. and longitude 58° 12 W. % 
sails on the following courses: 



BEARINGS. 

MILES. 

1 

S. 67° 30' E. 

12 

2 

s. 

6 

3 

S. 67° 30' W. 

6 

4 

N. 50° 374'E. 

32 


Required the latitude and longitude arrived at. 

Aim. Latitude, 37° 55' N.; Longitude, 57° 34' AY. 








346 


SURVEYING AND NAVIGATION. 


Example 10. 

A ship in latitude 43° 25' N., and longitude 65° 35' AY., sails 
on the following courses : 



BEARINGS. 

MILES. 

1 

S.AY.bS. 

63 

2 

S.S.AY.’AY. 

45 

3 

S.bE. 

54 

4 

S.AV.bAY. 

74 


Required her course and distance thence to Sandy Hook 
light, in latitude 40° 27^' N. and longitude 74° 11' AY. 

Ans. Course, N. 88° 14' AA r .; Distance, 276 miles. 


Example 11. 

A ship sails from latitude 50° 8' N. and longitude 4° 24' AY., 
on the following courses. 



BEARINGS. 

MILES. 

1 

S. 19° 41’' W. 

18 

2 

S. 75° 561' W. 

22 

3 

S. 53° 26J' AY. 

58 


Required the latitude and longitude arrived at. 

Ans. Lat. 49° 11' K; Lon. 6° 18' AY. 


Example 12. 

A ship from Cape Clear, latitude 51° 55' K, longitude 9° 
29' AY., sails as follows : 















NAVIGATION. 347 



BEARINGS. 

MILES. 

1 

S.bW. 

23 

2 

w.s.w. 

40 

3 

S.W.fW. 

18 

4 

W.iN. 

28 

5 

S.b.E. 

12 

6 

S.S.E.fE. 

16 


Required her course, and distance made good; also, latitude 
and longitude arrived at. 

Ans. Course, S. 45° 47' W .; Distance, 102.4 miles; Lati¬ 
tude, 50° 44' N.; Longitude, 11° 25' W. 


Example 13. 

A ship in latitude 41° 12' N., longitude 37° 21' W., sails as 
follows: 



BEARINGS. 

MILES. 

1 

S.W.bW. 

21 

2 

S.W.JS. 

31 

3 

W.S.W.JS. 

16 

4 

S.f E. 

18 

5 

S.W.JW. 

14 

0 

W.|N. 

30 


Required her course, distance, latitude and longitude. 

Ans. Course, S. 52° 40' W.; Distance, 111.7 miles; Lati¬ 
tude, 40° 5' ~N. ; Longitude, 39° 18' W. 


Example 14. 

Last noon we were in latitude 28° 40' S., and longitude 32° 
20' W. ; since then we have sailed by the log, 















348 


SURVEYING AND NAVIGATION. 


- 

BEARINGS. 

MILES, j 

1 

S.W.fW. 

62 

2 

S.bW. 

16 

3 

W.JS. 

40 

4 

S.W.fW. 

29 

5 

S.bE. 

30 

6 

S.JE. 

14 

! 


Required the direct course and distance, and our present 
latitude and longitude. 

Ans. Course, S. 43° 14' W.; Distance, 158 miles; Latitude, 
30° 41' S.; Longitude, 34° 24 f W. 


Example 15. 

A ship from Toulon,, latitude 43° V N., longitude 5° 56 1 E. 
sailed, 



BEARINGS. 

MILES. 

1 

s.s.w. 

48 

2 

S.bE. 

34 

3 

S.W.fW. 

26 

4 

E. 

17 


Required her course and distance to Port Mahon, latitude 
39° 52' H., and longitude 4° 18 f 30" E. 

Ans. Latitude of ship, 41° 32 f N.; Longitude, 5° 37' E. 
Course to Port Mahon, S. 30° 45f W. nearly ; and distance, 
116.4 miles. 


Example 16. 

On leaving the Cape of Good Hope for St. Helena, we took 
our departure; Cape Town bearing S. E. by S. 12 miles, 
after running N. W. 36 miles, and N. W. by W. 140 miles. 





















NAVIGATION. 


349 


Required our latitude and longitude, and tlie course and dis¬ 
tance made. 

N. B. Latitude of Cape Town, 33° 5G' S. Longitude, 1S° 
23' E. 

Latitude of St. Helena, 15° 55' S. Longitude, 5° 43' 30" W. 
Ans. Latitude, 32° 3' S.; Longitude, 15° 25' E.; Course, 
N. 52° 41' W .; Distance, 1ST miles. 


SECTION IY. 

MERCATOR’S SAILING. 

Mercator’s Sailing is a method of Li- 

determining difference of longitude 

derived from the following prin- g_ 

ciple. 

In the figure, let AC be distance, 

AB difference of latitude, and 
BC departure. Now, if we take 
AL greater than AB in the ratio 
of radius to the cosine of the lati¬ 
tude of BC , and draw LD parallel 
to BC\ it will represent the differ¬ 
ence of longitude for the two points 
A and C\ for 

LD : BC: : AL : AB. 

Therefore, 

LD : BC : : R : Cosine latitude. 

But from Proportion 1 we have 

Difference of longitude : Departure : : R : Cosine latitude. 







350 


SURVEYING AND NAVIGATION. 




Therefore, if BC is departure, LD must be difference of 
longitude. 

AL is called the meridional difference of latitude, to distin¬ 
guish it from AB , the proper difference of latitude. 

To find AL , the meridional difference of latitude, a table 
called a Table of Meridional Parts has been computed. If we 
take any small portion of the meridian AB , and divide it by 
the cosine of its latitude, or multiply it by the secant of its lati¬ 
tude, radius being unity, we shall get a corresponding portion 
of the increased meridian AL ; and taking a nautical mile as 
the unit of the table, beginning at the equator, we shall get 
secant P, secant 2', secant 3', secant 4', and so on, for the sev¬ 
eral minutes of the increased meridian. Adding these secants, 
we shall get the numbers called meridional parts for the lati¬ 
tudes 1', 2', 3' 4', and so on. 

Thus, Secant 1' 

Secant l'-)-secant 2' 

Secant l'-f secant 2'-f secant 3' 

Secant l'-f-secant 2'-j-secant 3'-f secant 4 f , etc., 
give the numbers of the table in nautical miles. These num¬ 
bers express the distance from the equator to the correspond¬ 
ing parallel of latitude, measured on the increased meridian. 
From this table we can take the meridional parts correspond¬ 
ing to each latitude, and by subtracting one number from the 
other when both latitudes are north or south, or by adding them 
when one is north and the other south, we shall obtain the 
meridional difference of latitude in numbers. 

In the figure we have 

1. AB : ALj :: BC : LD ; or 

The proper difference of latitude is to the meridional differ¬ 
ence of latitude as the departure is to the difference of longi¬ 
tude. 

2. B : tsin.BA C :: AL : LD ; or 

Radius is to the tangent of the course as the meridional 
difference of latitude is to the difference of longitude. 


v 


X 


NAVIGATION. 


35 X 


Example 1. 

A. ship from latitude 32° 30' N. and longitude 18° 20' W., 
makes the following courses and distances: 

1. N.W.bW. 150 miles. 

2. W.N.W. 124 “ 

3. S.S.W. 41 “ 

Required the latitude and longitude arrived at, and the 
course and distance made good. 

Take the latitudes and departures from the traverse table, 
and arrange them as in the tablet. 



COURSES. 

DIST. 

N. 

S. 

y 

E. 

W. 

1 

KW.bW. 

150 

83.3 



124.7 

2 

W.N.W. 

124 

47.5 



114.6 

3 

S.S.W. 

41 

% 

37.9 


15.7 




130.8 

37.9 


255.0 




37.9 





92.9 = diff. of latitude. 

Whence, 32° 30' N., latitude given, 

1° 33' N., difference of latitude. 

34° 3 ; N., latitude arrived at. 

To find the longitude, take from the table the meridiona 
parts for 34° 3 f , which will be 2175 

Also the parts for 32° 30', “ “ “ 2064 

Subtracting we get the meridional diff. of latitude = 111 

Then from the tablet, we see that the departure is 255 miles, 
the proper difference of latitude is 93 miles nearly ; and since 
proper difference of latitude is to meridional difference of lati- 

























352 SURVEYING AND NAVIGATION. 

tude, as is departure to tlie difference of longitude, we 
have 

Log. 255 = 2.406540 

Log. Ill = 2.045323 

4.451863 
Log. 93 = 1.968483 

Log. 304.4 = 2.483380 

which gives the difference of longitude equal to 304 nearly, 
or 5° 4 f west. 

And 18° 20' W., longitude given, 

5° 4' W., difference of longitude, 

23° 24' W., longitude arrived at. 

To find tlie course, we have 

Log. 255 - 2.406540 

Log. 93 — 1.968483 

Tan. 69° 5S' 10.438057 

Therefore the course is N. 69° 58' "W. 

To find the distance we have 

Log. 255 - 2.406540 

Sin. 69° 58' = 9.972894 

Log. 271.4 miles = 2.433646 

In determining difference of longitude from the proportion, 

Cos. lat. : 1 :: departure : diff. longitude, 

when the ship sails on an oblique course, we assume that 
the departure is the same as would have been made on that 
parallel, which is the arithmetical mean of the latitude 
left and arrived at. This assumption is not true. The mean 
latitude found in that way does not always give the true dif¬ 
ference of longitude. 


* 








NAVIGATION. 


353 


From the proportion, 

Diff. lat. : meridional diff. lat 
we get 


diff. lat._ 

meridional diff. lat. 


departure : diff Ion. 


departure 
diff Ion. 



If we put ip for the latitude of the parallel that will give 
the true longitude, we have 

Cos. xp : 1 :: departure : diff Ion. 


Therefore 

Cos. xp 

_ departure 

(2) 


diff Ion. 

from 1 and 2, 

we get 




Cos. i b — 

diff. lat. 

(3) 


meridional diff. lat. 


Hence, we see that the proper difference of latitude divided 
by the meridional difference of latitude will give the cosine of 
the required latitude, and this compared with the arithmetical 
mean of the given latitudes will give the correction to be 
added to the mean latitude. 


Example 2. 


Suppose the given latitudes are 30° and 40°, the proper dif¬ 
ference of latitude will be 10° or 600 miles; the meridional 
difference will be 

Meridional parts for 40° = 2622.7 

Meridional parts for 30° = 1888.4 

Meridional difference of lat. == 734.3 


Therefore, cos. ip = 


600 

734.3 


And Log. 600 = 2.778151 

Log. 734.3 = 2.865874 

Cos. 35° 12' = <1912277 


The mean of 40° and 30° is 35°, hence the correction 
is 12'. 


In this manner the following table may be computed : 











354 


SURVEYING AND NAVIGATION 


Table of corrections in minutes to be added to the arithmetical 
mean to find the true mean latitude . 


DIFFERENCE OF LATITUDE. 


MEAN 

LAT. 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

f 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18* 

19° 

to 

o 

o 


/ 

/ 

/ 

/ 

/ 

f 

r 

f 

/ 

f 

/ 

/ 

/ 

f 

r 

f 

/ 

/ 

/ 

15° 

1 

2 

3 

5 

7 

9 

12 

15 

18 

22 

26 

31 

86 

41 

47 

52 

59 

65 

7: \ 
69 

16° 

1 

2 

3 

4 

6 

9 

11 

14 

18 

21 

25 

30 

£4 

39 

44 

50 

56 

62 

17° 

1 

2 

3 

4 

6 

8 

11 

14 

17 

20 

24 

28 

33 

88 

43 

48 

54 

60 

06 

18° 

1 

1 

3 

4 

6 

8 

10 

13 

16 

20 

23 

27 

32 

86 

41 

46 

52 

58 

64 

19° 

1 

1 

3 

4 

6 

8 

10 

13 

16 

19 

22 

26 

30 

85 

40 

45 

50 

56 

61 

21° 

1 

1 

2 

4 

5 

7 

10 

12 

15 

18 

22 

25 

29 

84 

38 

43 

48 

54 

60 

23° 

1 

1 

2 

3 

5 

7 

9 

11 

14 

17 

20 

23 

27 

31 

35 

40 

45 

50 

6ft 

25° 

1 

1 

2 

3 

5 

7 

9 

11 

13 

16 

19 

23 

26 

80 

34 

39 

43 

48 

63 

30° 

I 

1 

2 

3 

5 

6 

8 

10 

13 

15 

18 

21 

25 

28 

82 

86 

41 

45 

60 

35° 

0 

1 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

82 

36 

40 

45 

49 

0 

O 

1 

1 

2 

3 

5 

6 

8 

10 

13 

15 

18 

21 

25 

28 

32 

36 

41 

45 

60 

45° 

1 

1 

2 

3 

5 

6 

S 

11 

13 

16 

19 

22 

26 

80 

34 

38 

43 

48 

53 

50° 

1 

1 

2 

4 

5 

7 

9 

11 

14 

17 

20 

24 

28 

82 

86 

41 

46 

51 

67 

55° 

1 

1 

2 

4 

6 

8 

10 

13 

16 

19 

22 

26 

31 

35 

40 

45 

51 

57 

63 i 

60° 

1 

2 

3 

4 

6 

9 

11 

14 

18 

22 

26 

80 

35 

40 

46 

52 

5S 

65 

72 

62° 

1 

2 

3 

5 

7 

9 

12 

15 

19 

23 

27 

32 

87 

43 

49 

55 

62 

70 

77 

64° 

1 

2 

3 

5 

7 

10 

13 

16 

20 

24 

29 

34 

40 

46 

52 

59 

67 

75 

83 

66° 

1 

2 

3 

5 

8 

11 

14 

18 

22 

26 

32 

87 

43 

50 

57 

64 

72 

81 

SO j 

/ 68° 

1 

2 

4 

6 

8 

12 

15 

19 

24 

29 

84 

40 

47 

54 

62 

70 

79 

89 

99 1 

70° 

1 

2 

4 

6 

9 

13 

16 

21 

26 

32 

38 

44 

52 

60 

68 

78 

88 

98 

110 I 

71° 

1 

2 

4 

7 

10 

13 

17 

22 

27 

83 

40 

47 

55 

63 

72 

82 

93 

104 

116 I 

72° 

1 

3 

5 

7 

10 

14 

IS 

23 

29 

35 

42 

49 

58 

67 

76 

87 

98 

111 

124 I 

73° 

1 

3 

5 

8 

11 1 

15 

19 

25 

31 

37 

44 

52 

61 | 

11 

81 

93 

105 

118 

188/ 


\ 




Examples. 


1. A ship from Cape Finisterre, in lat. 42° 56' FT., and 
longitude 8° 16' W., sailed S. W. \ W. till her difference of 
longitude is 134 miles ; required the distance sailed, and the 
latitude in. 


As radius 

: diff. Ion., 134 miles, 

:: cot. course, 4] points, 

: mer. diff. lat., 121.5 miles, 


10.000000 

2.127105 

9.957295 

2.084400 


Lat. Cape Finisterre 42° 56' N. ; Mer. parts 2858 

Mer. diff 121 

Lat. 41° 27' N., corresponding to 2737 in table. 

As cosine course 9.827085 

: proper diff. lat., 89 miles, 1.949390 
:: radius 10.000000 

: dist. 132.5 miles, 2.122305 


i 








































































NAVIGATION. 


355 


2. 1 ship from lat. 40° 41' H., Ion. 1G° 37' "W., sails in the 
2(. E. quarter till she arrives lat. 43° 57' N., and has made 
248 miles departure; required her course, distance, and longi¬ 
tude in. 

Ans. CourseN. 51° 41' E.; Dist. 316 miles; Lon. in 11° W. 

3. IIqw far must a ship sail N. E. 1 E. from lat. 44° 12' 1ST., 
Ion. 23 W., to reach the parallel of 47° K.; and what from 
that point will be the bearing and distance of ITshant, which 
is in lat. 48° 28' K, and Ion. 5° 3' W. ? 

Ans. She must sail 264.8 miles, and her course to Usliant 
will then be N. 80° 32' E., and distance 535 
miles. 

4. A ship from the Cape of Good Hope steers E. a S. 446 
miles ; required her place, and her course and distance to 
Kerguelen’s Land, in lat. 48° 41' S., and Ion. 69° E. 

Ans. Lat. 35° 13' S., Ion. 27° 21' E.; course to Kerguelen’s 
Land, S. 66° 25' E., and distance 2018 miles. 

5. By observation, a ship was found to be in lat. 41° 50 ' S., 
Ion. 68° 14' E. She then sailed N. E. 140 m., and E. a S. 
76 m. ; required her place, and her course and distance to the 
island of St. Paul, which is in lat. 38° 42' S., and in Ion. 
77° 18'E. Ans. Lat. 40° 18' S., Ion. 72° 6' E.; course to St. 

Paul H. 68° 15'E., and dist. 259 miles, nearly, 

DIRECT METHOD OF COMPUTING MERIDIONAL PARTS. 

This presentation of Mercator’s Sailing is thought sufficient 
for all practical purposes. But it may be desirable to have a 
method of computing the Table of Meridional Parts directly 
for any degree of latitude without the preceding computa¬ 
tions, especially if we wish to test the accuracy of any part of 
the tables. It may also be interesting to the mathematical 
student to examine the Theory of Mercator’s Sailing in a 
manner differing entirely from the preceding. 


356 


SURVEYING AND NAVIGATION. 


It will be seen in what follows, that from the definition of 
the rhumb line, we can derive an equation that will give the 
difference of longitude when we have the latitude left, the lati¬ 
tude arrived at, and the course known, without a table of 
meridional parts. 

Also, that from the hypothesis of Mercator’s Sailing, we can 
derive an equation that gives directly the meridional parts for 
any latitude. 

Let EQ be the equa¬ 
tor, and P its pole; let 
ASB be the rhumb line 
or track of a ship, sailing 
from A to B ; put C for 
the course of the ship, 
and let PS equal 0 , the 
co-latitude of the ship, 
and </> equal the angle APS, which is the longitude of the 
ship reckoned from the meridian passing through A. Then 
we shall have, 

dfismM = 0 (1) 

de ’ w 

wdience we get, 

7 , _ tan. CdS 
sin .0 

Integrating, we get, 

(p = tan. C log. tan.^ 0 . 

Taking the limits PA and PB , we get, 

<f> = tan.(7 (log. tan.^fl'—log. tan. 10") (3) 

where 0 f is the co-latitude of the place left, and 0 " is the co-lati 
tude of the place arrived at. The logarithms in equation (3) 
are Naperian, and if we use the common tables, the right hand 
member of (3) must be divided by the modulus of our sys- 








NAVIGATION. 


357 


tern, which is .4342944819. And to obtain <p in nautical miles, 
we must multiply by 155—55_, which will give, 

7T 

cp = 7915.7 tan .G (log. tan.^0'—log. tan.’#*') (4) 

Equation (4) gives the difference of longitude when the lati¬ 
tudes of the places, and the course, are given. 


Example. 


A ship in latitude 32° 38' N. and longitude 16° 55' AY., sails 
N. 79° 37' W. until she reaches the parallel of 40° 2' N. Re¬ 
quired the longitude arrived at. 

In equation (4), we must put C = 79° 37', 0' = 57° 22', 
and 0" = 49° 58', and we shall get 

(p = 7915.7 tan. 79° 37,' (log. tan. 28° 41'—log. tan. 24° 59'). 

Tan. 28° 41' = 9.738071 
Tan. 24° 59' = 9.668343 

.069728 


And, 


Log. .069728 = 2.843407 

Tan. 79° 37' = 10.736995 

Log. 7915.7 = 3.898489 


Log. 3012 


3.478891 


Therefore, 0 = 3012 miles, or 50° 12'. Hence the longi¬ 


tude arrived at is 67° 7' AY. 

To compute a table of meridional parts, we may put 0 for 
the latitude of any parallel, and 0 for the distance of the same 
parallel from the equator; then from Mercator’s theory, we 


shall have, 



d0 

cos.0 



The integral of this equation gives 

0 = log. tan. (45°-f-^0). (2) 


For common logarithms and nautical miles as the unit, 






358 


SURVEYING AND NAVIGATION. 


we must multiply tlie right hand number of (2) by 7915.7; 
whence we get, 

*0 = 7915.7 log. tan. ^45° + 



Example 1. 

Required the meridional parts for 20° of latitude. 

Here 0 = 20°, whence, 

ip = 7915.7 log. tan. 55°. 

Tan. 55° = 10.154773; and 

Log. .154773 = -1.189696 

Log. 7915.7 = 3.898489 

Log. 1225.1 = 3.088185 

Therefore 0 = 1225.1 miles, which is the number found in 
the table of meridional parts for 20°. 


Example 2. 


Required the meridional parts for latitude 78° 52’. 

Here 0 = 78° 52 f , and equation (3) becomes 

0 = 7915.7 log. tan. 84° 26'. 

Tan. 84° 26 f = 11.011158 ; and we have 

Log. 1.011158 = .004819 

Log. 7915.7 = 3.8.98489 

Log. 8004 = 3.903308 

Whence 0 = 8004 miles, the meridional parts for the given 
latitude, 78° 52'. 


i 




NAVIGATION. 


359 








SECTION y. 

SAILING IN CURRENTS. 

If a ship at B, sailing in the direction BA, were in a cur¬ 
rent which would carry her from B to C, in the same time 

that in still water she would sail from D 0 

B to A, then, by the joint action of 
the current and the wind, she would 
in the same time describe the diagonal 
BD of the parallelogram ABCD. For her being carried by 
the current in a direction parallel to BC, would neither alter 
the force of the wind, nor the p’osition of the ship, nor the 
sails, with respect to it; the wind would therefore continue to 
propel the ship in a direction parallel to AB, in the same man¬ 
ner as if the current did not exist. Hence, as she would be 
swept to the line CD by the independent action of the cur¬ 
rent, in the same time that she reached the line AD by the 
independent action of the wind on her sails, she would be 
found at D, the point of intersection of the lines AD and CD, 
having moved along the diagonal BD. 

Problems relating to the oblique action of a current upon a 
ship may be resolved by the solution of a plane triangle, as 
ABD in the preceding figure, where, if BA represent the dis' 
tance a ship would sail in still water, and AD the drift of 
the current in the same time, BD will be the actual dis- 
1 mce sailed, and ABD the change in the course produced by 
the current. 

EXAMPLES. 

1. If a ship sail W. 28 miles, in a current which in the same 
time carries her N. N. W. 8 miles, required her true course 
and distance. 





360 


SURVEYING AND NAVIGATION. 


Conceive the current to be one course and distance, and 
with the other course find the course and distance made good. 
Thus, by the traverse table: 


COURSE. 

DIST. 

DIFF. LATITUDE. 

DEPARTURE. 

N. 

8 . 

E. 

W. 

w. 

28 




28 

N.N.W. 

8 







7.39 



3.06 



7.39 



31.06 


7.39 : E : : 31.06 : tan. 76° 37', the course, 
Sin. 76° 37' : J2 : : 31.06 : 31.93, the distance. 


2. If a ship sails E. 7 miles an hour by the log, in a current 
setting E. N. E. 2.5 miles per hour, required her true course 
and hourly rate of sailing. 

Ans. Course, N. 84° 8' E. ; rate, 9.358 per hour. 

3. A ship has made by the reckoning 17. i W. 20 miles, but 

by observation it is found that, owing to a current, she has 
actually gone N. N. E. 28 miles. Required the setting and 
drift of the current in the time which the ship has been run¬ 
ning. Ans. Setting, N. 64° 48' E.; drift, 14.1 miles. 

4. A ship’s course to her port is W. N. W., and she is run¬ 
ning by the log 8 miles an hour ; but meeting with a current 
setting W. 1 S.4 miles an hour, what course must she steer in 
the current that her true course may be W. N. VV. ? 

Ans. Course,N. 53° 52' 2" YV. 

5. In a tide running VV. b W. 3 miles an hour, I wished 
to weather a point of land which bore N. E. 14 miles. What 
course must I steer so as to clear the point, the ship sailing 
7 miles an hour by the log, and what time shall I be in reach¬ 
ing the point ? 

Ans. Course, N. 69° 51' E.; time, 2 hours 25 minutes. 























N A VIGATION. 


301 


6. From a ship in a current, steering "W. S. W. 6 miles an 
hour by the log, a rock was seen at 6 in the evening, bearing 
S. W. ^ S. 20 miles. The ship was lost on the rock at 11 P. M. 
Required the setting and drift of the-current. 

Ans. Setting, S. 75° 10* E.; drift, 3.11 miles per hour. 

PARALLAX. 

To obtain the true altitude of a body from its apparent alti¬ 
tude, as given by an instrument, certain corrections are neces¬ 
sary. 

These corrections are for semi-diameter , dip, refraction and 
parallax. The correction for semi-diameter is obvious. 

At sea, the visible horizon (from which all observed alti¬ 
tudes are taken) is where the sea and sky apparently meet; 
and when the eye of the observer is above the water, this visi¬ 
ble horizon is below the sensible horizon, and the amount of 
the depression is called the dip of the horizon. Its correction 
is always subtractive, and its amount is to be found in Table V. 

Refraction is to be found in Table VII. It is always sub¬ 
tractive; for the reason, see some treatise on natural phil¬ 
osophy. 

Parallax is always additive. Conceive two lines drawn to a 
heavenly body, one from an observer at the circumference of 
the earth, and the other from the center of the earth. The in¬ 
clination of these two lines is parallax ; and when the body is 
in the horizon its parallax is greatest, and it is then called 
horizontal parallax. 

Parallax always tends to depress the object; but the parallax 
of any celestial object, except that of the moon, is so small, 
that we shall pay attention to lunar parallax only. This is 
so important to navigation that we shall give it a full explana¬ 
tion. 

The moon’s horizontal parallax is given in the Nautical Al¬ 
manac for every noon and midnight of Greenwich time, and 
from the horizontal parallax we must deduce the parallax 

1 1 


362 


SURVEYING AND NAVIGATION. 


corresponding to any other alti- 
, tude. 

Let AC he the radius of the 
earth, A the position of an ob¬ 
server, Z his zenith, and suppose 
H to be the moon in the hori¬ 
zon ; then the angle AHC is the 
moon’s horizontal parallax, and the angle AhC is the parab 
lax corresponding to the apparent altitude JiAH. Draw Am 
parallel to Ch ; then mAIl will be the true altitude . 

From this figure we draw the following definition for hori¬ 
zontal parallax. 

The horizontal parallax of any body is the angle under 
which the semi-diameter of the earth would appear as seen 
from that body. Of course, then, when the body is at a great 
distance, its horizontal parallax must be small; hence the sun 
and the remote planets have very little parallax, and the fixed 
stars none at all. 

Let CII and Ch be each represented by R. Put p — the 
horizontal parallax, and x — the parallax in altitude, or the 
angle mAh or AhC. 

Now in the triangle ACH , right-angled at A , we have, 

1 : sinjp \ \ R \ AC. 

In the triangle ACli , we have, 

sin. CAh : sin.# : : R : AC. 

By comparing these two proportions, we perceive that 

1 : sinjp : : sm.CAh : sin.#. 

Whence, sin.# = sin.jp sin. CAh. 

But sm.CAh — cos .hAH, for the sine of any arc greater 
than 90° is equal to the cosine of the excess over 90°; hence, 

sin.# = sinjp cos. hAII. 

The lunar horizontal parallax is rarely over a degree, com¬ 
monly less, and the sine of a degree does not materially differ 






NAVIGATION. 


363 


from the arc itself; hence, the preceding equation becomes, 
without any essential error the following : 

x — p cos. altitude. 

Or, in words, the parallax in altitude is equal to the hori¬ 
zontal parallax multiplied by the cosine of the apparent alti¬ 
tude (radius being unity). 


EXAMPLES. 

1. The apparent altitude of the moon’s center, after being 
corrected for dip and refraction, was 31° 25'; and its horizon¬ 
tal parallax at that time, taken from a nautical almanac, was 
57' 37" ; what was the correction for parallax, and what was 
the true altitude as seen from the center of the earth ? 

p = 57' 37" = 3457" ; log. 3457" = 3.538699 

cos. 31° 25* = 9.931152 

Log. 2950 = 49' 10" = x = 3.469851 
Ans. Cor. for parallax, 49' 10" ; true altitude, 32° 14' 10". 

2. The apparent altitude of the moon’s center on a certain 
occasion was 42° 17', and its horizontal parallax at the same 

time was 58' 12" ; what was the parallax in altitude, and what 

\ 

was the moon’s true altitude ? 

Ans. Parallax in altitude, 43' 4" ; true altitude, 43° 0' 4". 


SECTION YI. 

LATITUDE. 

1. To find the latitude of a place by the sun at noon. 

The latitude of a place may be determined from the meri¬ 
dian altitude of the sun. 

The altitude taken with a sextant must be corrected for 




364 


SURVEYING AND NAVIGATION. 


semi-diameter, parallax, dip, and refraction. The correction 
for semi-diameter is found in the almanac, and it must be sub¬ 
tracted or added, according as the upper or lower limb of the 
sun was observed. 

The correction for parallax is found by multiplying the 
sun’s parallax by the cosine of the observed altitude; it must 
always be added. 

The corrections for dip and refraction are found in tables, 
and must be subtracted to give the true altitude. 

Then from the true altitude subtract the sun's declination 
found in the almanac for that time, and the remainder will be 
the co-latitude. Or subtract the true altitude from 90°, and 
get the zenith distance. 

Then if the sun and observer are on the same side of the 
equator, to the zenith distance add the sun’s declination to 
obtain the latitude ; if the sun and observer are on different 
sides of the equator, from the zenith distance subtract the 
sun’s declination to obtain the latitude. 

EXAMPLES. 

1. On a certain day, the meridian* altitude of the sun’s lower 
limb was observed to be 31° 44', bearing south. At that time 
its declination was 7° 25' 8" south, semi-diameter 16' 9", index 
error+ 2' 12", height of the eye 17 feet. What was the lati¬ 
tude ? Ans. 50° 38 1 north. 

* To obtain the meridian altitude of the sun, the observer commences 
observations before noon, while the sun is still rising, driving the index for¬ 
ward as fast as the image appears to rise; and there will come a time, a few 
minutes in succession, in which the image appears to rest on the horizon, 
neither rising nor falling; but at length the image will fall; then the observer 
knows that noon has passed, and the greatest apparent altitude will be 
shown by reading the index. 


NAVIGATION. 


365 


Observed altitude = 31° 44' 0" 


Semi-diameter 

+ 

16' 

9" 


32° 

O' 

9" 

Index error 

= +2' 

12" 


32° 

2' 

21" 

Dip 

— _ 

- 4' 

4" 


31° 

58' 

17" 

Refraction 

~ — 

- 1' 

32" 


31° 

56' • 

45" 

Parallax 


4- 

7" 

True altitude 

- 31° 

56 1 

52" 

Zenith distance 

= 58° 

3' 

8” 

Declination 

= 7° 

25 1 

8" 

Latitude north 

= 50° 

38' 

0" 


2. The meridian altitude of the sun's uppe r limb was 40° 42* 
bearing north ; the declination was 23° 22' south ; semi¬ 
diameter was 16' 17" ; height of the observer was 16 feet. Re¬ 
quired the latitude. 

Corrections: 


Sun’s semi-diameter 

- — 

-16' 17" 

Dip 

= — 

- 3' 56" 

Refraction 

— 

- 1' 6" 

Parallax in altitude 

F- 

+ 

11 



-21' 10" 

Observed altitude of limb 

11 

o 

o 

12' 0" 

True altitude of sun’s center 

o 

O 

II 

20' 50" 

Zenith distance 

o 

ca 

II 

39’ 10" 

Declination 

= 23° 

22' 0' 

Latitude south 

II 

—i 

CO 

o 

1' 10" 


In this example the semi-diameter is subtracted because 
the upper limb was observed ; the dip and refraction are always 
subtracted, and parallax is added. 

3. The meridian altitude of the sun’s lower limb was 60° 2.V 















366 


SURVEYING AND NAVIGATION. 

' • ✓ 

bearing south, semi-diameter 16' 8", height ot the observer 
16 feet, and the sun’s declination 9° 22' north. What was 
the latitude ? 

Corrections: 


Sun’s semi-diameter 

— 

+ 16' 8" 

Dip 


- 3' 56" 

Refraction 

— 

- O' 33" 

Parallax in altitude 

=: 

+ 4" 

+ 11' 43" 

Observed altitude 

— 

60° 25' 0" 

True altitude of sun’s center 

— 

60° 36' 43" 

Zenith distance 


29° 23' 17" 

Declination 

— 

9° 22' 0" 

Latitude north 

— 

38° 45' 17" 


find the latitude from each of the following meridian 
observations: 



OW«CCT. 

ALT. OB. 

DIKKC. 

8. D. 

HEIGHT. 

DECLINATION. 

LATITUDE. 

1 

Sun L.L. 

45° 27' 

South 

16'15" 

20 feet 

17° 19' 31" S. 

27° 

2' 51" N. 

2 

Sun L.L. 

81° 43' 

-South 

15' 47" 

14 “ 

22° 13' 7" N. 

30° 

18' 10" N. 

3 

Suu L.L. 

87° 29' 

South 

16' 17" 

16 “ 

22° 9' S. 

19° 

50' 12" S. 

4 

Sub L.L. 

15° 45' 

South 

16' 0" 

16 “ 

4° 43' S. 

69 

23' 16" N. 


Navigators usually add 12' to the observed altitude of the 
srm’s lower limb as a correction for the sun’s semi-diameter, 
parallax, and dip; then subtract the refraction due to the 
altitude as given in the tables of refraction. The remainder 
will give the true altitude nearly, when the observation is 
taken upon the deck of a common sized vessel. 

The following table gives the correction for the sun’s semi¬ 
diameter, dip, and refraction, when the sun’s lower limb is ob¬ 
served. 






























NAVIGATION 


£67 


s 

M '4 

CO 5 

CORRECTION 

SUN’S 

TO BE ADDED TO 
LOWER LIMB TO 

TIIE OBSERVED ALTITUDE OF THE 
FIND TIIE TRUE ALTITUDE. 

bun's oi; 

AJLT1T 




HEIGHT OP THE EYE 

AliOVE THE SEA 

IN FEET. 




6 

8 

10 

12 

14 

16 

IS 

20 

22 

24 

26 

28 

30 

Ofl 

34 

1 

c 

/ 

9 

9 

9 

/ 

9 

/ 

9 

9 

t 

9 

9 

9 


J 

5 

3.S 

3.5 

3.1 

2.8 

2.5 

2.3 

2.1 

l.S 

1.6 

1.4 

1.2 

1.0 

o.s 

0.6 

0.51 

6 

5 3 

4.9 

4.6 

4.3 

4.0 

8.7 

8.5 

3.3 

3.0 

2.8 

2.6 

2.4 

2.2 

2.1 

1.9| 

7 

C.4 

6.0 

5.7 

54 

5.1 

4.8 

4.6 

4.4 

4.1 

3.9 

8.7 

3.5 

3.3 

3.2 

8.o\ 

S 

7.2 

6.8 

6.5 

6.2 

5.9 

5.7 

5.4 

5.3 

5.0 

4.S 

46 

4.1 

4 2 

4.0 

8.9 

9 

7.9 

7.5 

7.2 

6.9 

6.6 

6.4 

6.1 

5.9 

5.7 

5.5 

5.3 

5.1 

4.9 

4.7 

4.5 

10 

8.5 

8.1 

7 3 

7.5 

7.2 

6.9 

6.7 

6.5 

6.2 

6.0 

5.8 

5.6 

5.4 

5.3 

5.1 

11 

8.9 

8.6 

S.2 

7.9 

7.6 

7.4 

7.2 

6.9 

6 7 

6.5 

6 3 

6.1 

5.9 

5.7 

5.6 

1 i 

9.3 

9.0 

8.7 

8.3 

8. > 

7.8 

7.6 

7.3 

7.1 

6.9 

6,T 

6.5 

6.3 

6.2 

6.0 

I 1 

9.9 

9 6 

9.2 

8.9 

8.7 

8.4 

8.2 

7.9 

7.7 

7.5 

7 3 

7.1 

6.9 

6.8 

6.6 

1.1 

10.4 

10.1 

9.7 

9.4 

9.1 

8.9 

8.7 

8.4 

8.2 

8.0 

7.8 

7.0 

7.4 

7.2 

7.1 

13 

10.S 

1 1.4 

10.1 

918 

9.5 

9 3 

9.0 

8.S 

8.6 

8.4 

8.2 

8.0 

7.S 

7.6 

7.5 

20 

11.1 

1 i.7 

1 '.4 

1 ) 1 

9.8 

9.6 

9 3 

9.1 

8.9 

8.7 

8.5 

8.2 

8.1 

7.9 

7.7 

22 

11.4 

11.0 

10.7 

10.4 

10.1 

9.3 

9.6 

9.1 

9.1 

8.9 

8.7 

8.5 

8.3 

8.2 

8.0 

26 

11.7 

11.4 

11.0 

10.7 

lo.5 

10.2 

10.0 

9.7 

9.5 

9.3 

9.1 

8.9 

8.7 

8.6 

8.4 

30 

12.0 

11.7 

11.3 

11.0 

10.8 

1J.5 

10.3 

10.0 

9.8 

9.6 

9.4 

9.2 

9.0 

8.9 

8.7 

35 

12.3 

11.9 

11 6 

11.3 

11.0 

10.7 

10.6 

10.3 

10.1 

99 

9.7 

9.4 

9.2 

9.2 

9.0 

4) 

12.5 

12.2 

11. s 

11.5 

11.3 

11.0 

10.8 

10 5 

10.3 

10.1 

9.9 

9.7 

9.5 

9.4 

9.2 

45 

12.7 

12.4 

12.o 

11.7 

11.5 

11.2 

11.0 

10.7 

10.5 

10.2 

10.1 

9.8 

9.7 

9.6 

9.4 

50 

12.8 

12 5 

12.2 

11.9 

11.6 

11.3 

11.1 

10.9 

10.6 

10.4 

10.3 

10.0 

9.8 

9.7 

9.5 

55 

13.0 

12.6 

12.3 

12.0 

11.7 

11.5 

11.2 

11.0 

10.7 

10.5 

10.3 

10.1 

9.9 

9.8 

9.6 

60 

13.1 

127 

12.4 

12.1 

11.8 

11.6 

11.3 

11.1 

10.9 

10.6 

104 

10.2 

10.1 

99 

9.7 

65 

13.2 

12.8 

12.5 

12.2 

11.9 

11.7 

11.4 

11.2 

11.0 

10.7 

10.5 

lo..; 

KM 

10 0 

9.S 

70 

13.3 

12.9 

12.6 

12.3 

12.0 

11.8 

11.5 

11.3 

11.0 

10.8 

10.6 

10.4 

10.2 

10 1 

9.9 

75 

13.4 

13.1 

12.7 

12.4 

12.1 

119 

11.7 

114 

11.2 

11.0 

10.8 

1(1.6 

10.4 

10.2 

10.1 

80 

i 

13.6 

11.2 

12.9 

12.6 

12.3 

i2.0 

11.8 

11.6 | 

n. 

11.1 

10.9 

10.7 

10.5 

10.4 

10.2 


Monthly Correction 

Jan. 

-t0'.3 

Feb. 

+0'.2 

Mar. 

-to'.i 

April. 

O'.O 

May. 

—O'.l 

June. 

—0'.2 

for 

Sun’s Semi-di -.meter. 

July. 

—0'.3 

Aug. 

—052 

Sept. 

—O'.l 

Oct. 

-fO'.l 

Nov. 

-f0'.2 

Dec. 

+0'.3 


2. To find the latitude by 
the sun at any given time. 

Take the altitude of the 
sun, note the time, correct 
the observed altitude for dip, 
semi-diameter, and retrac¬ 
tion ; also, take the declina¬ 
tion from the Almanac. Then 
in the triangle ZPS, ZS will 
be the complement of the 


z 















































































368 


SURVEYING AND NAVIGATION. 


altitude, PS the complement of declination, and the time at 
which the observation was taken subtracted from 12*, if the 
observation was taken before noon, will measure the angle at 
P. Therefore, we shall have ZS and PS, and the angle P 9 to 
find PZ , which is the co-latitude. If we put 0 for the arc from 
P to where the perpendicular falls from S, we shall have 
cos .P = cotPS tan.0; from which we get 6. And we also 
have, 

Cos..PA: cos ZS : : cos.0 : cos .6\ 

where 0 l is the arc from Z to where the perpendicular meets 
PZ produced. The difference of these arcs gives PZ, the 
co-latitude. 


Example 1 . 


At 8 OT afternoon, the true altitude of the sun’s center was 
found to be 70° 58 f , the declination of the sun being 21° N. 
Kequired the latitude. 

In this example, ZS — 19° 2 1 

PS = 69° O' 

P = 2° O' 


If w r e form a right-angled triangle, with PS as hypotenuse 
and 0 for its base, we shall have cos. 2° = cot. 69° tan 6. 

Therefore, tan.0 = (OS l —; hence 6 = 68° 59* 18**. 

cot. 69° 

Again, we have cos. 69° : cos. 19° 2 ? : : cos.0 : cos.0*, where 
6 } is the base of the triangle whose hypotenuse is 19° 2'. 


Therefore, cos.0* = 


cos. 19° 2'* cos. 68° 59* 18" 

_ • 

cos. 69° 


hence 0' = 18° 56' 43". 

And, 68° 59' 18" 

. 18° 56' 43" 


PZ — 50° 2’ 35" 
Hence the latitude is 39° 5 V 25’’ 





NAVIGATION. 


3G9 


Example 2 . 

At 32 m before noon, the sun’s true altitude was found to be 
19° 41' bearing south, the sun’s declination being 20° south. 
What was the latitude of the place of observation ? 

32 minutes converted to arc gives 8° for the angle at P; 
then we shall have, 

P — 8° 

ZS = 70° 19' 

PS = - + 20° 

2 

Let 6 = the arc which is the base of the triangle whose 
hypotenuse is PS, and let O' be the base of the triangle whose 
hypotenuse is ZS; then we shall have, 

Cos. 8° = cot.^-f 2O°jtan.0. 

From which we get, 

Q = 110° 10' 49". 

Again, we have 

C os> ^_p20°^ : cos. 70° 19' : : cos. 110° 10' 49" : cos.0 f 
From which we get, 

O' = 70° 8' 21" 

But 6—6’ = PZ = co-latitude. 

Therefore, 110° 10' 49" 

70° 8' 21" 

Co-latitude = 40° 2' 28" 

Latitude = 49° 57' 32" 

This method does not give good results when the sun is far 
from the meridian at the time the altitude is taken. A small 
error in the angle at P occasions a large error in the co-lati¬ 
tude PZ. 



370 


SURVEYING AND NAVIGATION 


✓ 


SECTION YII. 

LONGITUDE. 

Longitude, from celestial observations, is measured by time. 
A place 15° west of another will have noon one hour of abso¬ 
lute time later; if 30° west, noon will be two hours later, 
Ac., &c. ; 15° corresponding always to an hour in time. 
Therefore, if we have any way of determining the times at 
two places, corresponding to the same absolute instant, the 
difference of such times will correspond to the difference of 
longitude between the two places, at the rate of 15° to an 
hour, or 4 minutes to a degree. 

Hence if we have a chronometer set to Greenwich mean 
time, and we know the rate of the chronometer, we can deter¬ 
mine the longitude of any place whose local time can be 
ascertained. The difference between the Greenwich and local 
times is the longitude of the place, which will be east or west, 
according as the local time is earlier or later than the Green¬ 
wich time, as shown by the chronometer. 

Having the latitude of a 
place, the local time can be 
found from an observed alti¬ 
tude of the sun ; the declina¬ 
tion of the sun being taken 
from the almanac. 

In the diagram, let Z be 
the zenith of the place, P 
the north pole, S the place 
of the sun when its altitude 
was observed. PZ will be 
co-latitude, PS co-declination, and ZS will be co-altitude. 
In the triangle PZS , we have all the sides to find the angle 
at P , which is the angular distance of the sun from the meri- 











NAVIGATION. 


371 


dian of the place where the altitude was taken. Hence, if we 
compute the angle ZPS, and convert it to time by allowing 
one hour for fifteen degrees, we shall have the time at which 
the altitude w r as taken. 

Let S = the half sum of the sides of the triangle PZS, then 
from Spherical Trigonometry, we have 

q os _i p _ / sin. Asin.(A— ZS) U 
' sm.PZsm.PS / 

Example 1. 

In latitude 39° 6' 20'' north, the sun’s decimal ion being 
12° 3' 10" north, the true altitude of the sun’s center was 
found to be 30° 10' 40" rising. What was the time ? 

In this example 

PZ = 50° 53' 40" 

PS = 77° 56' 50" 

ZS = 59° 49' 20" 

Hence, S = 94° 19' 55" . 

And S-ZS = 34° 30' 35" 

Therefore, Sin. S — 9.998758 

Sin .(S-ZS) = 9.7532 35 

19.751993 
Sin. PZ = 9.889853 

Sin. PS = 9.990319 

19.880172 

2)19.871821 

Cos. 1 P = cos. 30° 22' = 9.9359l0 

Therefore \ P — 30° 22' 

And P - 60° 44' 

This, converted to time, gives 4* 2 m 56 s , or the altitude was 
taken at 7 h 57 m 4* in the morning. This is local apparent time. 
The equation of time found in the almanac for the da’y on 
which the observation was taken will correct this for local 
mean time ; and the difference between local, and Greenwich, 
mean time, will be the longitude. 










372 


SURVEYING AND NAVIGATION. 


Example 2. 


In latitude 43° 30' north at, 7 A 43 m in the morning by the 
watch, the altitude of the sun’s lower limb was taken 
32° 4', the height of the observer being 16 feet, the sun’s 
declination being 19° 50' 47" north, semi-diameter 15' 49" 
the equation of time —3"* 51', and the chronometer indica¬ 
ting 9* 0 m 46' Greenwich mean time. What was the longb 
tude ? 


Observed altitude lower limb = 32° 4 ; 0" 
Semi-diameter = -f 15' 49" 

Dip = — 3' 56" 

Detraction = — 1* 30" 


True altitude of center = 32° 14' 23" 


1 


In this example we have 


Hence, 

And 


PZ 

PS 

zs 

s 

S-ZS 


— 46° 30' 0" 
= 70° 9' 13" 
= 57° 45' 37" 
= 87° 12' 25” 

= 29° 26' 48" 


Therefore, 

Sin.# = +9.999483 

Sin.(£— ZS)= +9.691623 

Sin .PZ = —9.860562 

Sin.-P# = —9.973408 

2)1 9785713 6 

Cos. 31° 58' 8" = 9.928568" = eos 


Therefore j P = 31° 58' 8" 
And P = 63° 56' 16" 



This value of P converted to time gives 4‘ 15“ 45*, which 
is the time before noon when the altitude was taken, which 






NAVIGATION. 


373 


gives 7 h 44 m 15* for the local apparent time; but since the 
equation of time is — S m 51*, we have 

7 h 44 m 15* 

—3 m 51* 

7* 40 ,n 24* = local mean time. 

Watch time = 7 h 43 m 0* 

2 m 36* watch too fast. 

Greenwich mean time by chronometer = 9* 0 m 46* 

Time by observation 7 h 40 m 24* 

1* 20 w 22* 

which is the longitude of the place of observation ; and it is 
west, the local time being behind Greenwich time. 


Example 3. 


In latitude 54° 12' north, at 5 h S3 m afternoon by the watch, 
the true altitude of the sun’s center was found to be 16° 26' 
52". The sun’s declination was 15° 26' 50" north, and the 
equation of time was -f5 m ; the chronometer indicated 7 h 12”* 
45*, Greenwich mean time. What was the error of the watch, 
and what was the longitude of the place of observation ? 

In this example, we have, 


PZ 

PS 

zs 

Hence we find S 
And S-ZS 


= 35° 48' 0" 
= 74° 33' 10" 
= 73° 33' 8" 

= 91° 57' 9" 
= 18° 24' 1" 


Sin.,# = + 9.999747 

Sin.(#-Z#) = + 9.499210 

Sin.P^ = - 9.767124 

Sin .PS = - 9.984021 

2)19.747812 


Cos. 41° 34' 55" 


9.873906 










374 


SURVEYING AND NAVIGATION. 


Hence, £ P = 41° 34' 55" 

And P = 83° 9' 50" 

This gives in time 5 A 32”* 39* 

Applying equation of time, +5 

5* 37 m 39* 

which is the local mean time. 

Greenwich mean time by chronometer = 1 h 12”* 45* 
Local time by observation = 5 A 37”* 39* 

l h 35”* 6* 

which is the longitude required. 

Local time by observation = 5* 37”* 39* 

Watch time = 5* 33 m 0* 

Error of watch = . 4”* 39* 

Example 4. 

In latitude 21° 36' north, the true altitude of the sun’s cen¬ 
ter was found to be 30° 24' 40", the sun’s declination being 
21° 41' 40" south, equation of time + 8 m 31*, chronometer 
indicating 6 A 0”* 21 s P. M. Greenwich mean time. Required 
the longitude of the place of observation. 

In this example, we have 



PZ = 

68 

© 

CM 

o 


PS = 

111 

0 41' 40" 


zs 

59 

° 35' 20" 

Therefore, 

s 

119 

° 50’ 30" 

And 

f 

% 

II 

60 

° 15' 10" 


Sin.$ = 

+ 

9.938221 


Bin.(S—ZS) = 

-b 

9.938631 


Sin .PZ 

— 

9.968379 


Sin.PS = 

— 

9.968094 



2) 19.940379 


Cos. 20° 59' 14" = 


9.970189 







Therefore, 

And 


NAVIGATION. 


375 


i P = 20° 59' 14" 

P = 41° 58' 28" 

AA liicli gives, 2 h 47 w 54* local apparent time. 

Equation of time, S m 31* 

2 h 5G m 25* local mean time 

Greer wich mean time = 6* 0 m 21* 

3 h 3 m 56* longitude west. 


Example 5. 


In latit ado 0° 41 f south, the true altitude of the sun’s center 
was found to be 32° 31' 52" rising, the declination of the sun 
21° 2' 36 1 ' south, the equation of time +9 m 53*; the chrono¬ 
meter indicated 11* 27 m 41* A. M., Greenwich mean time.' 
What was the longitude by chronometer? 


Here we have, 


Hence, 

And 


Therefore, 

And 


PZ 

PS 

ZS 

s 

S-ZS 

Sin.$ 

Sin .(S-ZS) 
Sin PZ 
Sin .PS 


= 89° 19' 0" 
.= 68° 57' 24" 
= 57° 28' 8" 

- 107° 52' 16" 
= 50° 24' 8" 

= -f 9.978522 
= + 9.886794 
= - 9.999969 
= - 9.970025 


2) 19.895322 


Cos. 27° 34' 4" = 9.947661 

%P = 27° 34' 4" 
P = 55° 8' 8" 






376 


SURVEYING AND NAVIGATION. 


This changed to time gives 
The time of the day will be 
Equation of time, 

Greenwich mean time, 


3* 40™ 33* 

8* 19™ 27* apparent time. 

+ 9™ 53* 

8 h 29 m 20* local mean time. 
11* 27™ 41 s 

2* 58™ 21* = Ion. required. 


SECTION YIII. 

LUNAR OBSERVATIONS. 

A good and well-tried chronometer is a valuable and relia¬ 
ble instrument for finding the longitude at sea, during short 
runs; but still it is but an instrument, and is not one of the 

0 • 

reliable works of nature. Near the end of a long voyage the 
best of chronometers very frequently give false longitude; and 
in such cases good navigators always resort to lunar observa¬ 
tions, which, from the hands of a good observer, can be relied 
upon to within 10 or 12 minutes of a degree. Indeed, they 
usually come within 5 or 6 miles, and sometimes even more 
exact; but that is accidental and unfrequent. 

To comprehend the theory of lunars, we must call to mind 
the fact that the moon moves through the heavens, apparently 
among the stars, at the rate of more than 13° in a day, and 
any angular distance it may have from the sun or any star 
corresponds to some moment of Greenwich time. 

About three days before and after the change of the moon, 
•she is too near the sun to be visible ; but at all other times her 
distance from the sun, some of the larger planets, and certain 
bright fixed stars, called lunar stars,* which lie near her path, 
are computed and put down in the nautical almanac, for every 
third hour of mean Greenwich time, commencing at noon. 

* There are nine lunar stars, Arietis, Aldebaran, Pollux, Regulus, Spica, 
Antares, Aquilae, Fomalhaut, and Pegasi. 





NAVIGATION. 


877 


For any particular day, the distances are given to such objects 
only, east and west of her, as will be convenient to measure 
with the common instruments. 

The distances put down in the nautical almanac are such as 
would be seen if viewed from the center of the earth; but ob¬ 
servers are always on the surface of the earth, and the dis¬ 
tances thence observed must always be reduced to equivalent 
distances seen from the center by the application of spherb 
cal trigonometry. This reduction is called working a lunar. 

The moon is never seen by an observer in its true place, un¬ 
less the observer is in a line between the center of the earth 
and the moon, that is, unless the moon is in the zenith of the 
observer; in all other positions the moon is depressed by par¬ 
allax, and appears nearer to those stars that are below her, 
and further from those stars that are above her, than she would 
appear from the center of the earth. Therefore the apparent 
altitudes of the two objects must be taken at the same time 
that their distance asunder is measured. The altitudes must 
be corrected for parallax and refraction, thus obtaining the 
true altitudes. 

FIRST METHOD OF WORKING A LUNAR. 




378 


SURVEYING AND NAVIGATION. 


The apparent altitudes subtracted from 90° degrees give 
ZS 1 and ZmJ, and /S'm 1 is the apparent distance ; with these 
three sides the angle Z can be found. Correcting the altitudes 
and subtracting them from 90° will give the sides Z/S and 
Zm\ these two sides, and their included angle at Z, will ghe 
the side Sni which is the true distance. 

The definite true distance must have a definite Greenwich 
time, which can readily be found; and this, compared with 
the local time deduced from an altitude of the sun, will give 
the longitude. 

Let A — the apparent altitude of a star, 
and A = the true altitude. 

Let m l = the apparent altitude of the moon, 
and m — the true altitude. 

Also let d = the apparent distance of the moon from 

the star, 

and x = the true distance. 

Then from the fundamental equations of Spherical Trigo¬ 
nometry, we have the following, 


Also 

Whence, 


Cos. Z = 


Cos. Z = 


cos A —sin. A'sin.772/ 
cos. A cos .m! 

cos.a?—sin. A sin.771 


cos. A cos.m 
Cos. 6?—sin. A sin .W cos.a:—sin.Asin. 77 i 


COS. A COS.'771 


/ ' — 


COS. A COS.771 


By adding unity to each member, we have 

^ cos.rZ—sin.A sin.777 f _ 1 + cos.x—sin.Asin. 77 i 
cos. A 7 cos JU 1 . cos.Acos. 77 i 

(Cos.A cos. 777/ — s in.A sin. 777/) -f cos.d_ 
cos. A cos .m' 

(cos.Aco s. 777 —sin. A sin. 771) -f cos.a? 
cos. A cos. 771 


0 














NAVIGATION. 


379 


By observing equation 9, Plane Trigonometry, we perceive 
that the preceding equation reduces to 


Cos.j/S 7 + W) -f- co s, d cos.(^4-m)q-cos.£c 
cos.A' cos .m 1 

Whence 


cos.N cos.m 


Cos.x = (cos .(/S' -f m’) + cos^)- cos ^ cos,m ■— cos.(N+ m) (A) 

'cos ,/S 1 cos.m' 


It is here important to notice that the moon’s horizontal 
parallax given in the Nautical Almanac, is the equatorial 
horizontal parallax; that is, it corresponds to the greatest 
radius of the earth. The diameter of the earth through any 
other latitude is less, and of course the corresponding parallax 
is less. 

We therefore give the following table for the reduction of 
the equatorial horizontal parallax, to the horizontal parallax 
of any other latitude; it is computed on the supposition that 
the equatorial diameter is to its polar as 230 to 229. For 
example, if the horizontal parallax in the Nautical Almanac is 
55', in the latitude 40° the reduction would be 6", and the 
parallax reduced would be 54 f 54"; and if the parallax from 
the Nautical Almanac were GO' the reduction would be 6.6 f , 
and the reduced parallax would be 59' 53.4". 

The semi-diameter of the moon given in the Nautical 
Almanac is her horizontal semi-diameter ; but when she is in 
the zenith she is nearer to us by the whole radius of the earth, 
about one-sixtieth part of her whole distance. Consequently 
she must appear under a larger and larger angle as she rises 
from the horizon, and this is called the augmentation of the 
semi-diameter. 

We give the reduction for the parallax ; and the augmenta¬ 
tion for the semi-diameter in the following tables 





380 


SURVEYING AND NAVIGATION. 


RED. 

of moon’s eq. hor. 

PARALLAX. 


AUG. OF THE MOON’S 
SEMI-DIAM. 

Lat. 

Eq. Par. 

Eq. Par. 


Ap. Alt. 

Aug. 


55 ; 

60' 








o 

if 

o 

II 

II 


6 

2 

20 

0.9 

1 


12 

’ 3 

25 

2.8 

3 


18 

5 

30 

3.7 

4 


24 

6 

35 

4.6 

5 


30 

8 

40 

6.0 

6.6 


36 

9 

45 

7.3 

8 


42 

11 

50 

8.6 

9.4 


48 

12 

55 

10.1 

11 


54 

13 

60 

11 

12 


60 

14 

65 

11.8 

13 


66 

15 

70 

12.8 

14 


72 

16 

75 

13.9 

15 


90 

16 

80 

14.6 

16 





Example 1. 

On the 25th of Jan. 1852, between three and four o’clock 
in the afternoon, local time, the observed distance between 
the nearest limbs of the sun and moon was 50° 3' 20 ,f ; the 
altitude of the sun’s lower limb was 20° 1', and the altitude 
of the moon’s lower limb was 48° 57', height of the eye 16 
feet. The latitude corrected for the sun from noon was 34° 12' 
north, and the supposed longitude about 65° west. What .vas 
the longitude ? 


Preparation . 

Supposed time at ship = 3* 15 m P. M. 
Supposed longitude = 65° 4* 20 m 


Supposed Greenwich time = 7 h 35 m P. M. 




















NAVIGATION. 


381 


From the Almanac we find the 


Moon’s semi-diam. 
Aug. for Alt. 

Moon’s semi-diam. 


14' 46" ; Moon’s eq. lior. par. = 54' 12" 
12"; Red. for lat. = 4" 

14' 58"; Reduced hor. par. = 54' 8" 


Observed distance 
Sun’s semi-diameter 
Moon’s semi-diameter 


= 50 c 


3' 20" 
16' 16" 
14' 58" 


Apparent center distance = 50° 34' 34" 

Altitude moon’s lower limb = 48° 57' 00" 

Moon’s semi-diameter = 14' 58" 

Dip . = —3' 56" 

% ____________________ 

Moon’s apparent altitude = 49° 8' 2" = m! 
Alt. O’s lower L. = 20° 1' 


Semi-diameter 

Dip 

G’s app. alt. 
Refraction 

O’s true alt. 


= +16'16" 

= — 3' 56" 

= 20 ° ldT2Q"=ff 
= - 2 f 3 4" 

= 20° 10' 46" & 


N.B. To find the moon’s parallax in altitude see rule on 
page 363. 

d’s app. alt. 49° 8' cos. 9.815778 
54' 8" = 3248' log. 3.511616 

®’b par. in alt. = 35' 25" = 2125" 3.327394 

(t’s app. alt. = 49° 8' 2" 

Parallax in alt. = 35' 25" 

Refraction = — 49" 


True alt. = 49° 42' 38" = m 

(tf+m') = 69° 21 f 22" ; = 69° 53' 24" 


We are now prepared to apply the equation to compute the 










382 


SURVEYING AND NAVIGATION. 


true distance. The equation requires the use of natural sines 
and cosines. 


l / on . i\ . A COS ,/S COS .971 / rv . \ 

Cos.a? = (cos.hS'-l- 7 ft') + cos.tf) - - - f —cos.(o + m) 

v 1 cos.A cos.m' 


(S'+m l )= 69° 21 ' 22 "; N. cos. = .35256 
d = 50° 34' 34" ; N. cos* = .J33505 

.98761, log. =1994585 
S = 20° 10' 46", log. cos. = 9.972489 
m = 49° 42' 38", log. cos. = 9.810669 
/S’ = 20° IS' 20 ", cos. com. = 0.027631 
m' = 49° 8 ' 2 ", cos. com. = 0.184227 

Num. .97616, log. = 1.989521 

NT. cos.^+ra), 69° 53' 24", -.34382 
True distance, 50° 46' 37", cos.23463 

In the Nautical Almanac, we find that at 6 P.M. mean 
Greenwich time, on said day, the true distance between the 
sun and moon was 49° 59'26", and at 9 P.M. the distance was 
51° 20 ' 49 ", showing a change of 1° 2 P 23" in three hours of 
time. But the change from 49° 59' 26" to 50° 46' 37" is 47' 11"; 
and on the supposition that the change is proportional to the 
time, we have the following 

1° 2 P 23" : 47' 11" :: 3A : t 
Or 4883 : 2831 :: 3 h\ l h 48™ 22 * 


That is, the time that this observation was taken was 
1 * 44 ™ 22 * after 6 at Greenwich, or 7 h 44 m 22 * Greenwich 
mean time. 

With the true altitude of the sun 20 ° 10 ' 46", the latitude 


* When d is greater than 90° its cosine becomes minus , and its numeri¬ 
cal value is then the natural sine of the excess over 90°. Thus if d were 
105°, its cosine would be numerically equal to the sine of 15°, and must 
then be subtracted from the cosine of the sum of apparent altitudes. The 
result (cos x ) would then be the sine of the excess over 90°. 

t Less 20 because the table of natural sines is to radius unity, and we 
used cos. S and cos.m to the radius of 10, making two tens to take away. 





NAYIGATI ON. 


385 


34° 12', and the polar distance 109° O' 48", we find the 
apparent time at ship 3* l() m 5*, to which we would add the 
equation of time, 12™ 34 s , making the mean time 3 A 22 m 39 s . 

From Greenwich time = 7 A 44™ 22* 

Subtract time at ship = 3 22 39 

4* 21™ 43* = 65° 25' 45" Ion. W. 

This converted to degrees gives the longitude 65° 25' 45" 
west of Greenwich. 


Example 2. 


The apparent distance between the centers of the snn and 
moon was 98° 12', 

The sun’s apparent altitude was 54° 10', 

The moon’s altitude 20° 37', 

Moon’s parallax 57' 12", 

What was the true distance from the center of the sun to 
the center of the moon ? 

Moon’s apparent altitude = 20° 37' 

Refraction —2' 30" 

Parallax = 53' 32" 

Moon’s true altitude = 21° 27' 58" 


Sun’s apparent altitude = 54° 10' 0" 
Refraction = — 40" 

Sun’s true altitude = 54° 9' 20" 


{8+m') = 74° 47'; (S+m) = 75° 37' 18* 

Cos. 74° 47' = .26247 

Cos.fccos. 98° 12' = -14263 

Cos^xS 7 + m 1 ) -f cos.t/ = .11984 


>■ 






384 


SURVEYING AND NAVIGATION 


Log. .11984 

— 

1.078602 

Cos.,# 

= cos. 54° 9 f 20" = 

9.767591 

Cos.??z 

= cos. 21° 2 7 58" = 

9.968779 

Com. cos./S 7 

= com. cos. 54° 10' = 

0.232525 

Com. cos.?/?/ 

= com. cos. 20° 37' = 

0.028744 

Log. .11919 

— 

L076241 


+ .11919 

-Cos.OS'+m) = -cos. 75° 37' 18" = -.24833 
Cos.# = cos. 97° 25' 12" = -.12914 

which is the true distance. 


SECOND METHOD OF WORKING A LUNAR. 


The foregoing method of clearing a lunar distance is very 
good, as an educational exercise ; but for practical use, it is ob¬ 
jectionable, as the equation requires the use of natural sines 
and cosines. To insure a complete understanding of this im¬ 
portant subject, theoretically and practically, we will further 
transform the equation, 


/ \ cos.xS' cos .?n 

Cos.x = ^ cos. (o' + m!) + cos .dj cog g cog —cos.(tf+m), (1) 


and adapt it to the use of logarithmic sines and cosines. 

Conceiving ($' + ?/?/) to be a single arc, and applying equa¬ 
tion (17) (page 83), to the first factor in the second member of 
(1), we shall have, 


Cos.# = 

2 cos.^(^ + /??/ + ?£) cos.t! (S 1 -\-m j — <T) cos.aS" cos.?;?, 

cos.aS 7 cos.?/?/ 



— COS.(aS'+???). 


By equation (32) (p. 85), we find that cos.# = 1 — 2 sin. 2 /#. 
By equation (31) (p. 85), cos.^+y/z) = 2 cos. 2 ^ (£+?/?)— 1. 
These values of cos.# and cos.(xS'+??z), placed in (2) will give 

1—2 sin n# = 9 cos.^/S 7 + m l + d) cos , i +m'— d)vos.S cos.m 

cos.iS" cos.m' 

+1—2 cos. 2 ^($+??z). 









NAVIGATION. 


385 


By dropping unity from each member, and dividing by 
— 2 , we have, 


Sin . 2 lx = 
Cos. 2 ! (S-l-my 


cos4 


+ d) cos.^S 1 + 171 *-—d) cos.S cos. 
cos .JS 1 COS.m 7 


(3) 

.m. 


. j 


By division, we obtain. 

Sin. 2 ia; _ 

Cos. 2 1 ($ 4 - in) 

_Cos.4(A f 4- ni 1 + d) cos.^S 1 4 -ntf—d) vos.S cos .m , 

cos. 2 2 (^ 4 -m) cos./S" cos jrri) 

Assume 

Sin 2 P _ cos.:J(/S ff -\-m l + d) cos.^(aS"4 -nd — d) cos. $ cos.m 

cos. 2 2(^4- fn.) cos. A' cos .m 1 

Where P is an auxiliary arc; equation (4) now becomes 

__= 1 —sin. 2 jP. 

cos . 2 2 (o4- m) 

Since sin. 2 P+ cos. 2 jP — 1 , we liave cos . 2 P = 1 —sin. 2 P. 


Whence, 


sin. 2 4a; 2 n 

2 = COS. P. 


cos. 2 ^ (S-\-?n) 

By extracting square root and clearing of fractions, we have 

'\i 1 

Sin. 2 # = cos.P cos. 2 (S 4 -m). ( 6 ) 

Equations (5) and ( 6 ) are plain and practical ; they can bo 
easily remembered, and they are adapted to logarithms. 

Equations (5) and ( 6 ) can be put in words, and called a rule, 
but in our opinion this is not necessary. 

We will now re-eompute the last example, in which 

S* = 54° 10 7 , S = 54° 9' 20", m 7 = 20° 37', m = 21° 27 53", 

- . • d = 98° 12 f , {S' + m') = 74° 47'* 

17 








- / 


SURVEYING AND NAVIGATION. 



= 86° 29' 30", cos. 

(less 10) 

= 2.786704 

Md-S-m 1 ) 

= 11° 42' 30", cos. 

u 

= 1.990869 

s 

= 54° 9' 20", cos. 

a 

= T767591 

m 

= 21° 27' 58", cos. 

a 

= T968779 

l(S+m) * 

= 37° 48' 39", cos. 

complement 

= 0.102352 


*cos. 

complement 

= 0.102352 

S' 

= 54° 10', cos. 

complement 

= 0.232525 

in! 

= 20° 37', ‘ cos. 

complement 

= 0.028744 




2)2.979916 




— 1.490038 


Add 


10 


Sin.i 3 

, 17° 59' 56", 

= 9.489958 

Cos .P = 9.978209 



CoM(fl+m) = 9.897648 

Sin.^a? = 9.875857 - 48° 42' 35" 
Whence, true distance = 97° 23' 10", Ans. 

The two methods do not give the same result precisely. But 
this one is the more reliable of the two. 


EXAMPLES. 


NO. 

APPARENT ALT. OP 
SUN OR A FIXEO 8TAR. 

moon’s ap. 

ALTITUDE. 

APPARENT CEN¬ 
TRAL DISTANCE. 

MOON’8 
IIOR. PAR. 

TRUE 

DISTANCE. 



0 

9 

0 

f 

o 

/ 

>i 

/ 

// 

o 

f 

H 

1 

o 

86 

3 

39 

18 

46 

45 

0 

53 

51 

46 

4 

25 

2 

X 

29 

47 

67 

22 

27 

35 

0 

60 

3 

28 

8 

24 

3 

o 

31 

14 

28 

7 

14 

21 

30 

64 

29 

14 

9 

24 

4 

o 

60 

5 

63 

12 

61 

3 

21 

58 

30 

50 

41 

16 

5 

X 

34 

28 

10 

42 

49 

18 

38 

61 

11 

48 

45 

39 

6 

o 

8 

26 

19 

24 

120 

18 

46 

57 

14 

120 

1 

46 

7 

X 

43 

27 

40 

9 

18 

21 

35 

60 

20 

18 

8 

12 

8 

X 

53 

13 

67 

32 

60 

13 

49 

60 

52 

59 

48 

12 

9 

o 

72 

26 

18 

30 

81 

2 

28 

60 

58 

80 

9 

33 

10 

o 

60 

33 

9 

26 

70 

36 

16 

59 

57 

69 

49 

12 


* The preceding log. repeated to obtain the square of the last quantity. 


\ 






















NAVIGATION. 


387 


THIRD METHOD OF WORKING A LUNAR. 



Let Z be the ob¬ 
server’s zenith, 8 1 the 
apparent place of the 
sun or star, and 8 its 
true place ; m 1 the ap¬ 
parent place of the 
moon, m its true place. 

Then 8?n- will he the 
apparent distance of 
the moon from the sun 
or star, and 8m will he 
the true distance. 

Now if we regard the differences between these apparent 
and true places as -differentials, we shall obtain another for¬ 
mula for correcting Lunar Distances. 


Let 8 = the apparent altitude of the sun or star. 

m = the apparent altitude of the moon. 
Andcc= the apparent or observed distance, #V. 


Now by Spherical Trigonometry, (see page 136 of this 
hook), we have 


Cos .Z 


cos.#—sin.# sin.m 
cos.# cos .in. 



In this problem the angle Z is always a constant quantity, 
# and m are variable, and x varies in consequence of the 
variations of # and m. But we may take these effects separ¬ 
ately, that is, by supposing m only to vary, and discover the 
corresponding variation for x. Then we may suppose 8 to 
vary, and obtain the corresponding variation of x; and lastly, 
these two effects put together will be the total variation for x y 
or the difference between the apparent and true distance 
between the sun and the moon, or a star and the moon as the 
ease may be. 





388 


SURVEYING AND NAVIGATION. 


We will therefore differentiate (1) on the supposition that x 
and m are variables. Thus, 

.-•A 

cl. cos .Z cos./^ cos.m — cl. cos .x—d. sin.# sin.m. 

<• ■» 

Or —eoAZcos./S r simm*^m = —sin .x’dx —sin.# cos.m’dm. 

Sin.#^— = cos.Z cos.S sin.m—sin.# cos.m. (2) 

/ dm 

/ t 

But equation (1) gives 

Cos.Zcos.# = 

7 

Multiply by sin.m, then 


cos.#—sin. # sin.m 
v cos.m 


Cos.Zcos.# sin.m = 


cos.# sin.m—sin.# sin. 2 m 


COS.?/i 


This value placed in equation (2), that equation becomes 

gi n ;r ^_ cos.a:8in.m-sin./S , sin^m _ a;ii i<?f , f(g m 

dm cos.m 


Or 

Cos. 772 - sin.#-^ = cos.# sin.m—sin.#sin. 2 m—sin.# cos. 2 m. 
dm 

— cos.# sin.m—siii.#(sin. 2 m-pcos. 2 m). 

But (Sin. 2 m + cos. 2 m) = 1. Therefore 

. dx • • n! 

Cos.m sin.#— = cos.# sin.m—sm.o. 

dm 

Or dx = ( <**■* )dm (3) 

V cos.m sin.# / 

Now if we suppose that x and # are the variables, in place 
of x and m, the result will be the same as (3) if we change # 
to m and m to #. 


Therefore ( CQS,a? , s '^? l \dS is the value of dx corres- 

ponding to the variation of the sun’s or star’s latitude. 

The apparent place of the moon is below’ its true place, and 
the apparent place of the sun or star is above its true place, 
therefore dm and dS must have contrary signs. Consequently, 













389 


NAVIGATION. 


the whole variation of x , when both S and m vary, (as is 
always the case,) must be 

cos.a? sin.m—sin./SV^ /cos.a? sin.^—sin.?n 

cos.m sin.a? / \ cos .S sin.a? 

When the sun or star is at the zenith, dS is then nothing, 
and the value of dx is expressed by the first term of the sec 
ond member. When the moon is in the zenith, then dm 

\ ... i 

becomes nothing ; but in practice, such cases would not be 
likely to occur, once in a life time. 

We will work the fourth example by this tormula: 




Given the sun's apparent altitude 60° 5 ; . The moon's appa¬ 
rent altitude 63° 12'. The apparent distance 51° 3' 21", and 
the moon's horizontal parallax 58' 30", to find the true dis¬ 
tance from center to center , as seen from the center of the 
earth .* Ans. 50° 41' 15". 

In this example S = 60° 5 r ; m — 63° 12 f ; x — 51° 3' 21". 
r Moon’s h. p. = 58' 30" = 3510" 

Log. 3510 = 3.545307 

Cos. 63° 12 1 = 9.654059 

_■_«*» 

* ' Log. 1583 = 3J99366 

Therefore parallax in altitude = 1583" 

Refraction* == —29 

dm = 1554" , 

; ‘ * -?± —33", sun s refraction. 

\ . r 

For the coefficient of dm , 

Sin. m = 9.950650 cos. m = 9.654059 

Cos. x - 9.798348 sin. a? = 9.890845 

i . . ii. ■ i ■■ ‘t 1 _i 

Log. .56105 = L748998 ' ■- 1.54-1904 

Sin. S = -86675 

T *: • y - * 

C 

* Correction may be made for the figure of the earth by correcting the 
parallax for the latitude of the observer. See table preceding. 








890 


SURVEYING AND NAVIGATION. 


Therefore 


.56105 

-.86675 


1.544904 


—.30570 therefore log. .30570 = 1.485295 

T940391 

And log. dm — log. 1554 = 3.191451 

whence, log. 1355 = 3.131842 

Therefore -1355” 

\ cos.m sin.x / 

For the coefficient dS, 

Sin.A = 9.937895 cos.A = 9.697874 

Cos.cc = 9.798348 sin.cc = 9.890845 


—1.736243=.54481 
Subtract sin.m=.89259 


1.588719 


Therefore 


.34778; log. .34778 = 1.541305 

T7952586 

Log. dS = log. —33 = 1.518514 

Log. 29.6" 
cos.x sin.S —sin.m 


( 


l \dS 


cos. S sin .x 

Whence, putting these parts together, we find 

dx =-1355” + 29.6" = -22' 5" 

Apparent distance = 51° 3' 21" 

True distance = 50° 41' 16" 

The equation 


= 1.471100 
29.6" 


^cos.rcsin.m—sin.$) 

mm — 1 

( cos.cc sin.xS'— sin.m^ 

v cos.m sin.cc ) 


l cos. A sin.aj / 


can be put under another form. 

Assume Cos.cc sin.m = sin. A 

Cos.cc sin.$ = sin.i? 

Then we shall have, 

d x —l ^n.A —sin.A\^_/sin.i?—sin.mW^ 
' cos.m sin.cc / \ eos.Asin.cc / 


















391 


NAVIGATION. 

The first term of the second member will be positive or 
negative, according as A is greater or less than S. The 
co-efficient of dS is positive when B is greater than m; but 
dS is always negative; hence the product will be positive or 
negative, as m is greater or less than B. 

From trigonometry, we get by substituting for the differ¬ 
ence of the sines their values in factors, 

a dx = 

cos.^(^t + /S r ) siii.Jfyl ~/V)'/«(. cossin.|(j B~m)dS 

cos ,m sin.# Cos./S' sin.# 


Example. 


Suppose that in latitude 46° north, the moon’s apparent 
altitude was 36° 28 f , and that of a planet 24° 43 f , and their 
apparent distance asunder was 71° 46' 24". The moon's 
h. parallax at that time was 58 f 31", and that of the planet 29". 
What w r as the true distance of the moon from the planet ? 


Moon’s parallax = 58 f 31" 

Correction for lat. = 7.7" 


Log. 3503.3 

Cos. ait. = cos. 36° 28 f 
Log. 2817" 

Moon’s parallax in alt. 
Refraction 

dm 


58' 23.3’’ = 3503.3 

= 3.544477 
== 9.905366 

= 3.449843 

= 2817" 

= 77" 

= 2740" 


Planet. 

Log. 29” = 1.462398 

Cos. 24° 43' = 9.958271 


Parallax in alt. = -f 26.3" = 1.420669 
Refraction = —124" 

dS - 97.7" 









392 


SURVEYING AND NAVIGATION. 


Here m = 36° 28', S = 24° 43’, and a; = 71° 46'. 

For the auxiliary arcs A and B, 

Cos.*, 71° 46’, 9.495388 9.495388 

Sin.rn, 36° 28’, 9.774046 Sin.#, 24° 43', 9.621313 

A = 10° 43’, 9.269434 B - 7° 31' 9.116701 
S = 24° 43’ m = 36° 28’ 

KS+.4) =17° 43' l(m + B) = 21° 59' 

i{S—A) = 7° O' — B) = 14° 28' 

We now follow the formula. 

1st Term. 

■s • ^ • 

Cos4(iS'+J.) = cos. 17° 43' = 9.978898 

A) - sin. 7° 0’ = 9.085894 
Cos. com .m — cos. com. 36° 28' = 0.094634 
Sin. com.* = sin. com. 71° 46' = 0.022372 
Log. dm = log. 2740" = 3.437751 

—416.4" 2.619549 

2d Termv 

Cos = cos. 21° 59 f = 9.96721T 

Sin.i(m—^) = si n. 14° 28* = 9.397631 

Cos. com.,# — cos.com. 24°43 r = 0.041729 

Sin. com.a? = sin. com. 71° 46' = 0.022372 

Log. dS = log. — 97.7" = 1.989895 

-b26.2 fr 1.418834 

Tlie first term is minus because A is less than S, and the 
second term is plus, because it contains the product of two 
minus factors, (sin.7?—sin.?/*,) and dS. 

V * <y -** 

Whence, \dx =—416.4”+ 26.2’’ ='-390.2” 

Or, (lx = -780” = —13' 

Apparent distance = 71° 46 r 24^ 

True distance — 71° 33' 24 f * 










NAVIGATION. 




We give the following examples of distances between the 
moon and planets, for exercises: 


NO. 

1 

moon’s appa¬ 
rent ALT. 

planet's 

APPAR. ALT. 

MOON’S DI8T. 
FROM PLANET. 

moon’s hor. 

PARALLAX. 

PLANF.T’8 

PARALLAX. 

TRITE 

DISTANCE. 

; 

O f 

O f 

o t n 

1 '! 

It 

© III 

l 

58 36 

16 23 

69 37 20 

56 0 

31 

69 40 30 

2 

80 4 

35 30 

60 4 3 

61 16 

18 

59 58 57 

3 

16 26 

29 41 

98 15 31 

60 35 

30 

97 45 4 

4 

50 14 

51 3 

40 0 0 

54 50 

25 

39 44 42 

5 

62 12 

38 27 

37 50 34 

55 13 

23 

37 58 14 


























&, L. E. GURLEY’S PRICE LIST. 



r B> y, N, Y, s April 1st, 1866. 


♦♦♦ 


In common with all other manufacturers, we have been compelled by the great 
advance in the cost of labor, the war tax, and the materials used, to increase our 
old established prices for Instruments, &c. 

We believe, however, that in most cases thoy are still far below those of other 
makers of established reputation. 


SURVEYORS’ COMPASSES. 

Plain, with Jacob’s Staff Mountings, 4 inch Needle. $30 00 

“ “ “ “ “ 5 “ “ . 38 00 

** “ “ “ “ 6 “ “ . 42 00 

Vernier, with Jacob’s Staff Mountings, 6 inch Needle. 53 00 

Railroad, “ “ u “ “ “ . 83 00 

EXTRAS. 

Compass Tripod, with Cherry Legs. $8 00 

“ “ “ Leveling Screws and Clamp and Tangent Movement 18 00 

“ “ “ “ “ without “ “ “ “ 16 00 

“ “ “ Mountings, without Legs. 7 00 

Compound Tangent Ball. 7 00 

Brass Cover for Compass Glass.. 1 75 

Outkeeper, for keeping Tally... I 75 


TRANSIT INSTRUMENTS. 


*Vernier, Flair 

T.lcscope, 6 inch Needle, with Compass Tripod. 


•Surveyors’ * 

it 

4 “ 

(4 

“ Adjusting Tripod. 


00 


it 

5 “ 

(( 

u tt n 


00 

It u 

u 

5J “ 

u 

U U It 

... 165 

00 

Engineers’ “ 

it 

4 “ 

4 

U ti tt 


00 

u tt 

44 

6 “ 

44 

U U it 

... 185 

00 

u u 

U 

6 “ 

44 

with Watch Telescope. 

... 225 

00 

u it 

u 

5 “ 

44 

with Theodolite Axis. 


00 

u u 

44 

6 “ 

U 

with Two Telescopes. 


00 


• A M plain” telescope Is one without any of the attachments or extras, as we term them, 
such as the clamp and tangent, vertical c ,*ele and leveL 























FBIOE LIST 


EXTRAS. 

Vertical Circle, 3^ inch diameter, Vernier Reading to five minutes. $9 00 

“ “ 4^ 44 44 u a i n gi e minutes. 15 00 

Clamp and Tangent Movement to Axis of Telescope. 8 00 

Level on Telescope, with Ground Bubble and Scale. 15 00 

Rack and Pinion Movement to Eye Glass. 5 00 

Sights on Telescopes, with Folding Joints. 8 00 

Sights on Standards at right angles to Telescope. 8 00 


SC LAE, COMPASSES. 

Solar Compass, with Adjusting Sockets and Tripod.$215 00 

Solar Telescope Compass, “ “ “ “ . 240 00 

Micrometer Telescope, 16 to 20 inches long, with Rack Movement to Ob¬ 
ject Glass, and with Movable Clips to attach the Sights to No. 1. 28 00 


LEVELING RODS. 


Yankee or Boston. . $18 00 

New York, with Improved Mountings... 18 00 


LEVELING INSTRUMENTS. 


Sixteen inch Telescope, with Adjusting Tripod.$135 0(1 

Eighteen “ “ “ 44 “ 135 00 

Twenty “ “ “ 44 “ 135 00 

Twenty-two inch “ 44 “ 14 135 00 

Builders’ Level, with Adjusting Tripod. 50 00 


POCKET COMPASS ES. 

With Folding Sights, inch Needle, very serviceable for tracing lines 

once surveyed. $9 oo 

With Folding Sights, 2| inch Needle, with Jacob Staff Mountings. 11 50 

“ “ “ 3 ^ “ 44 “ “ “ “ . 13 50 

“ “ 3^ 44 44 without Jacob Staff Mountings. 11 00 

Without Sights, 1 to 2 inch Needle.from 25 cents to 6 00 

M ners’ Compass, or Dipping Needle, for tracing Iron Ore, a new and beau- 
tiful article, glass on both aides 


10 00 

























INFORMATION TO PURCHASERS. 


Manual. —To those who may wish to purchase any of the instruments 
mentioned in the previous pages of this Advertisement, we will send our 
Manual—a book of 125 pages, containing a full description of the same, 
with the adjustments, etc., free of charge (postage included), on applica¬ 
tion to us at Troy, N. Y. 

Instruments Wanted. —In regard to the best kind of instruments 
for particular purposes, we would here say, that where only common 
surveying, or the bearing of lines in the surveys for County Maps is 
required, a Plain Compass is all that is necessary. In cases where the 
variation of the needle is to be allowed, as in retracing the lines of an 
old survey, etc., the Vernier Compass, or the Vernier Transit, is re- 
• 

Where, in addition to the variation of the needle, horizontal angles 
are to be taken, and in cases of local attraction, the Railroad Compass 
is preferable; and for a mixed practice of Surveying and Engineering, 
we consider the Surveyor’s Transit superior to any instrument made by 
us or any other manufacturers. 

In the surveys of U. S. public lands, the county and township lines 
are required to be run by such instruments as the Solar Compass. 

Where Engineering is the exclusive design, the Engineers’ Transit 
and the Leveling Instruments are of course indispensable. 

Warranty.—A ll our instruments are examined and tested by us in 
person, and are sent to the purchaser adjusted and ready for immediate 
use. 

They are warranted correct in all their parts—we agreeing in the 
event of any defect appearing after reasonable use, to repair, or replace 
with a new and perfect instrument, promptly and at our own cost, ex¬ 
press charges included, or we will refund the money, and the express 
charges paid by the purchaser. 

Trial of Instruments. —It may often happen that this statement of 
the prices and quality of our instruments may come into the hands of 
those who are entirely unacquainted with us, or with the quality of our 
work, and who therefore feel unwilling to make a final purchase or 5 an 
article, of the excellence of which they are not perfectly assured. 

To such we make the following proposition : We will send the in¬ 
strument to the express station nearest the person givir-g the order, and 





398 


INFORMATION TO PURCHASERS. 






direct the Express Agent, on delivery of the same, to collect our bill, 
together with charges of transportation, and hold the money on deposit 
until the purchaser shall have had—say two weeks’ actual trial of its 
quality. 

If not found as represented, he may return the instrument before the 
expiration of that time, and receive the money paid, in full, including 
express charges, and direct the instrument to be returned to us. 

Packing, etc. —Each instrument is packed in a well-finished maho¬ 
gany case, furnished with lock and key and brass hooks, the larger ones 
having besides these a leather strap for convenience in carrying. Each 
case is provided with screw-drivers, adjusting-pin, and wrench for centre- 
pin, aud, if accompanied by a tripod, with a brass plumb-bob; with all 
instruments for taking angles, without the needle, a reading microscope 
is also furnished. 

Means of Transportation. —Instruments can be sent by express to 
almost every town in the United States and Canadas, regular agents 
being located at all the more important points, by whom they are for¬ 
warded to smaller places by stage. The charges of transportation from 
Troy to the purchaser are in all cases to be borne by him, we guaran¬ 
teeing the safe arrival of our instruments to the extent of express trans¬ 
portation, and holding the Express Companies responsible to us for all 
losses or damages by the way. 

Terms of Payment are uniformly cash, and we have but one price. 
Our prices for instruments are nearly one-third less than those of other 
makers of established reputation. They are as low as we think instru¬ 
ments of equal quality can be made, and will not be varied from the list 
given on the previous pages. 

Remittances may be made by a draft, payable to our order at Troy, 
Albany, New York, Boston, or Philadelphia, which can be procured 
from Banks or Bankers in almost all of the larger villages. 

These may be sent by mail with the order for the instrument, and if 
lost or stolen on the route, can be replaced by a duplicate draft, obtained 
as before, and without additional cost. 

Or the customer may pay the bill on receipt of the instrument to the 
Express Agent, taking care to send funds bankable in New York or 
Boston. The cost of returning bills collected by express of amounts 
under $1C will be charged to the customer. 

W. & L. E. GURLEY, 

Mathematical Instrument Makers, 

Fulton-st., opposite North End of Union R. R. Depot, Troy, N. Y 


— 


LOGARITHMIC TABLES; 

ALSO A TABLE 07 THE 

TRIGONOMETRICAL LINES; 


AND OTHER NECESSARY TABLES. 






LOGARITHMS OF NUMBERS 

non 

1 TO 10000 . 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0 000000 

26 

1 414973 

61 

1 707570 

76 

1 880814 

2 

0 301030 

27 

1 431364 

52 

1 716003 

77 

1 886491 

3 

ft 477121 

28 

1 447158 

63 

1 724276 

78 

1 892095 

4 

0 602060 

29 

1 462398 

64 

1 732394 

79 

1 897627 

5 

0 698970 

30 

1 477121 

65 

1 740363 

80 

1 903090 

6 

0 778161 

31 

1 491362 

66 

1 748188 

81 

1 908485 

7 

0 845098 

32 

1 605150 

67 

1 765876 

82 

1 913814 

8 

0 903090 

33 

1 618514 

68 

1 763428 

83 

1 919078 

9 

0 954243 

34 

1 631479 

69 

1 770852 

84 

1 924270 

10 

V 1 000000 

36 

1 644068 

60 

1 778151 

86 

1 929410 

11 

1 041393 

36 

1 656303 

61 

1 785330 

86 

1 934498 

12 

1 079181 

37 

1 668202 

62 

1 792392 

87 

1 939519 

13 

1 113943 

38 

1 679784 

63 

1 799341 

88 

1 944483 

14 

1 146128 

39 

1 591065 

64 

1 806180 

89 

1 949390 

16 

1 176091 

40 

1 602060 

65 

1 812913 

90 

1 954243 

16 

1 204120 

41 

1 612784 

66 

1 819544 

91 

1 959041 

17 

1 230449 

42 

1 623249 

67 

1 826076 

92 

1 963788 

18 

1 266273 

43 

1 633468 

68 

1 832509 

93 

1 968483 

19 

1 278764 

44 

1 643453 

69 

1 838849 

94 

1 973128 

20 

1 301030 

45 

1 663213 

70 

1 845098 

95 

1 977724 

21 

1 322219 

46 

1 662758 

71 

1 851258 

96 

1 982271 

22 

1 342423 

47 

1 672098 

72 

1 857333 

97 

1 986772 

23 

1 361728 

48- 

1 681241 

73 

1 863323 

98 

1 991226 

24 

1 380211 

49 

1 690196 

74 

1 869232 

99 

1 995636 

26 

1 397940 

50 

1 698970 

76 

1 876061 

100 

2 000000 


Note. In the following table, in the last nine columns of each page, where 
the first or leading figures change from 9’s to 0’s, points or dots are now 
introduced instead of the 0’s through the rest of the line, to catch tne eye, 
and to indicate that from thence the corresponding natural number in 
the first column stands in the next lower line f and its annexed first two 
figures of the Logarithms in the second column. 


> 























LOGARITHMS OF NUMBERS. 3 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2698 

3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

102 

8600 

9026 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2415 

103 

012837 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.361 

.775 

105 

021189 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

106 

6306 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

108 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

2676 

2969 

3362 

3755 

4148 

4540 

4932 

111 

6323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

112 

9218 

9606 

9993 

.380 

.766 

1153 

1638 

1924 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

6378 

5760 

6142 

6524 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9186 

9563 

9942 

.320 

115 

060698 

1075 

1452 

1829 

2206 

2682 

2958 

3333 

3709 

4083 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9668 

. .38 

.407 

.776 

1145 

1514 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

119 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

123 

9905 

.268 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6662 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

..26 

126 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

6169 

6510 

6851 

6191 

6531 

6871 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.263 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

130 

3943 

4277 

4611 

4944 

5278 

6611 

5943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

8265 

8595 

8926 

9256 

9586 

9915 

0245 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2644 

2871 

3198 

3525 

133 

3852 

4178 

4504 

4830 

6156 

6481 

6806 

6131 

6456 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9690 

..12 

135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

136 

3639 

3858 

4177 

4496 

4814 

6133 

6461 

5769 

6086 

6403 

137 

6721 

7037 

7354 

7671 

7987 

8303 

8618 

8934 

9249 

9564 

138 

9879 

.194 

.608 

.822 

1136 

1450 

1763 

2076 

2389 

2702 

139 

143015 

3327 

3630 

3951 

4263 

4574 

4885 

6196 

6507 

6818 

140 

6128 

6438 

6748 

7058 

7367 

7676 

7985 

8294 

8603 

8911 

141 

9219 

9527 

9835 

.142 

.449 

.756 

1063 

1370 

1676 

1982 

142 

152288 

2594 

2900 

3205 

3510 

3816 

4120 

4424 

4728 

5032 

143 

5336 

5640 

6943 

6246 

6649 

6862 

7164 

7457 

7769 

8061 

144 

8362 

8664 

8965 

9266 

9567 

9868 

.168 

.469 

.769 

1068 

145 

161368 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3768 

4056 

146 

4353 

4650 

4947 

6244 

6541 

6838 

6134 

6430 

6726 

7022 

147 

7317 

7613 

7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

148 

170262 

0555 

0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

149 

3186 

3478 

3769 

4060 

4361 

4641 

4932 

5222 

6612 

6802 
-. J 





































4 LOGARITHMS 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

160 

176091 

6381 

6670 

6959 

7248 

7636 

7825 

8113 

8401 

8689 

151 

8977 

9264 

9552 

9839 

.126 

.413 

.699 

.986 

1272 

1558 

162 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

163 

4691 

4975 

5259 

5542 

6825 

6108 

6391 

6674 

6956 

7239. 

164 

7621 

7803 

8084 

8366 

8647 

281 

8928 

9209 

9490 

9771 

..61 

165 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

156 

3125 

3403 

3681 

3969 

4237 

4514 

4792 

6069 

6346 

6623 

157 

6899 

6176 

6453 

6729 

7005 

7281 

7656 

7832 

8107 

8382 

158 

8657 

8932 

9206 

9481 

9755 

. .29 

.303 

.577 

.850 

1124 

159 

201397 

1670 

1943 

2216 

2488 

273 

2761 

3033 

3305 

3577 

3848 | 

160 

4120 

4391 

4663 

4934 

5204 

5475 

6746 

6016 

6286 

6666 

161 

6826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

162 

9515 

9783 

. .61 

.319 

.686 

.853 

1121 

1388 

1654 

1921 

163 

212188 

2454 

2720 

2986 

3252 

3618 

3783 

4049 

4314 

4579 

164 

4844 

6109 

6373 

6638 

6902 

264 

6166 

6430 

6694 

6957 

7221 

165 

7484 

7747 

8010 

8273 

8636 

8798 

9060 

9323 

9586 

9846 

166 

220108 

0370 

0631 

0892 

1153 

1414 

1676 

1936 

2196 

2456 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

6051 

168 

6309 

5568 

6526 

6084 

6342 

6600 

6868 

7116 

7372 

7630 

169 

7887 

8144 

8400 

8657 

8913 

257 

9170 

9426 

9682 

9938 

.193 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742 

171 

2996 

3250 

3504 

3767 

4011 

4264 

4517 

4770 

6023 

6276 

172 

6528 

6781 

6033 

6286 

6637 

6789 

7041 

7292 

7644 

7796 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

. .60 

.300 

174 

240549 

0799 

1048 

1297 

1546 

249 

1795 

2044 

2293 

2541 

2790 

175 

3038 

3286 

3634 

3782 

4030 

4277 

4525 

4772 

6019 

6266 

176 

6513 

6769 

6006 

6262 

6499 

6745 

6991 

7237 

7482 

7728 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

178 

250420 

0664 

0908 

1161 

1395 

1638 

1881 

2125 

2368 

2610 

179 

2853 

3096 

3338 

3580 

3822 

242 

4064 

4306 

4548 

4790 

6031 

180 

6273 

5514 

6765 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

7918 

8168 

8398 

8637 

8877 

9116 

9355 

9694 

9833 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

183 

2451 

2688 

2926 

3162 

3399 

3686 

3873 

4109 

4346 

4582 

184 

4818 

6054 

6290 

6526 

6761 

235 

6996 

6232 

6467 

6702 

6937 

185 

7172 

7406 

7641 

7876 

8110 

8344 

8578 

8812 

9046 

9279 

186 

9613 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

18 7 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

188 

4158 

4389 

4620 

4860 

6081 

6311 

6542 

6772 

6002 

6232 

189 

6462 

6692 

6921 

7151 

7380 

229 

7609 

7838 

8067 

8296 

8525 

190 

8764 

8982 

9211 

9439 

9667 

9895 

.123 

.361 

.678 

.806 

191 

281033 

1261 

1488 

1716 

1942 

2169 

2396 

2622 

2849 

3076 

192 

3301 

3527 

3763 

3979 

4205 

4431 

4666 

4882 

6107 

6332 

193 

6557 

6782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7678 

194 

7802 

8026 

8249 

8473 

8696 

224 

8920 

9143 

9366 

9589 

9812 

195 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1691 

1813 

2034 

196 

2258 

2478 

2699 

2920 

3141 

3363 

3684 

3804 

4026 

4246 

197 

4466 

4687 

4907 

6127 

6347 

6567 

5787 

6007 

6226 

6446 

198 

6666 

6884 

7104 

7323 

7642 

7761 

7979 

8198 

8416 

8636 

199 

8853 

9071 

9289 

9507 

9725 

9943 

.161 

.378 

.595 

.813 

































1 



OF NUMBERS 

• 



5 

1 N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

6136 

202 

6351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

203 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

204 

9630 

9843 

. .66 

.268 

.481 

212 

.693 

.906 

1118 

1330 

1542 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

6760 

207 

6970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

209 

320146 

0354 

0562 

0769 

0977 

207 

1184 

1391 

1598 

1805 

2012 

210 

2219 

2426 

2633 

2839 

3046 

3262 

3458 

3685 

3871 

4077 

211 

4282 

4488 

4694 

4899 

6105 

6310 

5516 

6721 

6926 

6131 

212 

6336 

6541 

6745 

6950 

7155 

7369 

7563 

7767 

7972 

8176 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

.. .8 

.211 

214 

330414 

0617 

0819 

1022 

1225 

202 

1427 

1630 

1832 

2034 

2236 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4061 

4253 

216 

4454 

4655 

4856 

6057 

5257 

6458 

6658 

6859 

6059 

6260 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

218 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

. .47 

.246 

219 

340444 

0642 

0841 

1039 

1237 

198 

1435 

1632 

1830 

2028 

2225 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

221 

4392 

4589 

4785 

4981 

6178 

6374 

5570 

6766 

6982 

6157 

222 

6353 

6549 

6744 

6939 

7135 

7330 

7625 

7720 

7915 

8110 

223 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

. .54 

224 

350248 

<H42 

0636 

0829 

1023 

193 

1216 

1410 

1603 

1796 

1989 

225 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

226 

4108 

4301 

4493 

4685 

4876 

6068 

5260 

6452 

6643 

5834 

227 

6026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7654 

7744 

228 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

229 

9836 

..26 

.215 

.404 

.693 

190 

.783 

.972 

1161 

1350 

1639 

230 

361728 

1917 

2105 

2294 

2482 

2671 

2859 

3048 

3236 

3424 

231 

3612 

3800 

3988 

4176 

4363 

4651 

4739 

4926 

5113 

5301 

232 

6488 

6675 

6862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

233 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030 

234 

9216 

9401 

9587 

9772 

9958 

185 

.143 

.328 

.513 

.698 

.883 

235 

371068 

1253 

1437 

1622 

1806 

1991 

2175 

2360 

2544 

2728 

236 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

237 

4748 

4932 

5115 

6298 

5481 

6664 

6846 

6029 

6212 

6394 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

239 

8398 

8580 

8761 

8943 

9124 

182 

9306 

9487 

9668 

9849 

..30 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296 

1476 

1666 

1837 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

242 

3816 

3995 

4174 

4353 

4633 

4712 

4891 

6070 

5249 

6428 

243 

6606 

6785 

6964 

6142 

6321 

6499 

6677 

6856 

■;034 

7212 

244 

7390 

7668 

7746 

7923 

8101 

178 

8279 

8456 

8634 

8811 

8989 

245 

9166 

9343 

9520 

9698 

9875 

. .61 

.228 

.405 

.582 

.759 

246 

390935 

1112 

1288 

1464 

1641 

1817 

1993 

2169 

2345 

2521 

247 

2697 

2873 

3048 

3224 

3400 

3576 

3761 

3926 

4101 

4277 

248 

4452 

4627 

4802 

4977 

6152 

6326 

6501 

6676 

6860 

6025 

249 

6199 

6374 

6648 

6722 

6896 

7071 

7245 

7419 

7592 

7766 































6 LOGARITHMS 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

397940 

8114 

8287 

8461 

8634 

8808 

8981 

9164 

9328 

9501 

251 

9674 

9847 

. .20 

.192 

.365 

.538 

.711 

.883 

1056 

1228 

262 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

254 

4834 

6005 

6176 

5346 

5617 

171 

6688 

5858 

6029 

6199 

6370 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 , 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9596 

9764 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 ! 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

259 

3300 

3467 

3636 

3803 

3970 

4137 

4305 

4472 

4639 

4806 

260 

4973 

5140 

6307 

5474 

5641 

5808 

5974 

6141 

6308 

6474 

261 

6641 

6807 

6973 

7139 

7306 

7472 

7638 

7804 

7970 

8135 

262 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

263 

9956 

.121 

.286 

.451 

.616 

.781 

.945 

1110 

1276 

1439 

264 

421604 

1788 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

265 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

266 

4882 

6045 

5208 

£371 

£634 

5697 

6860 

6023 

6186 

6349 

267 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

268 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

269 

9752 

9914 

..76 

.236 

.398 

.559 

.720 

.881 

1042 

1203 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

271 

2969 

3130 

3290 

3450 

3610 

3770 

3930 

4090 

4249 

4409 

272 

4569 

4729 

4888 

6048 

5207 

5367 

6526 

6685 

5844 

6004 

273 

6163 

6322 

6481 

6640 

6800 

6957 

7116 

7276 

7433 

7592 

274 

7751 

7909 

8067 

8226 

8384 

158 

8542 

8701 

8859 

9017 

9175 

275 

9333 

9491 

9648 

9806 

9964 

.122 

.279 

.437 

.694 

.752 

275 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

277 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

278 

4045 

4201 

4357 

4513 

4669 

4826 

4981 

6137 

6293 

6449 

279 

6604 

5760 

5916 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

280 

7158 

7313 

7468 

7623 

7778 

7933 

8088 

8242 

8397 

8562 

281 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

. .95 

282 

450249 

0403 

0557 

0711 

0865 

1018 

1172 

1326 

1479 

1633 

283 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

284 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4640 

4692 

235 

4845 

4997 

5160 

5302 

5454 

5606 

5768 

5910 

6062 

6214 

286 

6366 

6518 

6670 

6821 

6973 

7125 

",276 

7428 

7579 

7731 

287 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

288 

9392 

9543 

9694 

9845 

9995 

.146 

.296 

.417 

.697 

.748 

289 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

290 

2398 

2C48 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

291 

3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

6086 

5234 

292 

5383 

5532 

5680 

5829 

6977 

6126 

6274 

6423 

6671 

6719 

293 

6868 

7016 

7164 

7312 

7460 

7608 

7766 

7904 

8052 

8200 

294 

8347 

8495 

8643 

8790 

8938 

147 

9086 

9233 

9380 

9627 

9676 

295 

9822 

9969 

.116 

.263 

.410 

.657 

.704 

.851 

.998 

1145 

296 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

297 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

298 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

6381 

6526 

299 

__ 

6671 

5*16 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 







































OF NUMBERS. 7 


N. 

f 

0 

l 

2 

3 

4 

6 

6 

7 

8 

1 9 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

8278 

8422 

301 

8566 

8711 

8855 

8999 

9143 

9287 

9481 

9575 

9719 

9863 

302 

480007 

0151 

0294 

0438 

0582 

0725 

0869 

1012 

1156 

1299 

303 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2446 

2588 

2731 

304 

2874 

3016 

3159 

3302 

3445 

142 

3587 

3730 

3872 

4015 

4157 

305 

4300 

4442 

4585 

4727 

4869 

5011 

6153 

5296 

5437 

6579 

306 

6721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

337 

7138 

7280 

7421 

7563 

7701 

7845 

7986 

8127 

8269 

8410 

308 

8551 

8692 

8833 

8974 

9114 

9255 

9396 

9537 

9677 

9818 

309 

9959 

. .99 

.239 

.380 

.620 

.661 

.801 

.941 

1081 

1222 

310 

491362 

1502 

1642 

1782 

1922 

2062 

220i 

2341 

2481 

2621 

311 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4016 

312 

4155 

4294 

4433 

4572 

4711 

4860 

4989 

6128 

5267 

6406 

313 

5544 

6683 

6822 

6960 

6099 

6238 

6376 

b615 

6663 

6791 

314 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

315 

8311 

8448 

8686 

8724 

8862 

8999 

9137 

9275 

9412 

9650 

316 

9687 

9824 

9962 

. .99 

.236 

.374 

.611 

.648 

.786 

.922 

317 

501059 

1196 

1333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

318 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3618 

3655 

319 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

6014 

320 

6150 

6286 

6421 

6557 

6693 

6828 

6964 

6099 

6234 

6370 

321 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

322 

7856 

7991 

8126 

8260 

8396 

8630 

8664 

8799 

8934 

9008 

323 

9203 

9337 

9471 

9606 

9740 

9874 

. .9 

.143 

.277 

.411 

324 

510546 

0679 

0813 

0947 

1081 

134 

1215 

1349 

1482 

1616 

1760 

326 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

3084 

326 

3218 

3351 

3484 

3617 

3750 

3883 

401 o 

4149 

4282 

4414 

327 

4548 

4681 

4813 

4946 

5079 

5211 

5344 

6476 

6609 

5741 

328 

6874 

6006 

6139 

6271 

6403 

6535 

6668 

6800 

6932 

7064 

329 

7196 

7328 

7460 

7692 

7724 

7865 

7987 

8119 

8261 

8382 

330 

8514 

8646 

8777 

8909 

9040 

9171 

9303 

9434 

9666 

9697 

331 

9828 

9959 

. .90 

.221 

.353 

.484 

.615 

.745 

.876 

1007 

332 

521138 

1269 

1400 

1630 

1661 

1792 

1922 

2053 

2183 

2314 

333 

2444 

2575 

2705 

2835 

2966 

3096 

3226 

3356 

3486 

3616 

334 

3746 

3876 

4006 

4136 

4266 

4396 

4626 

4656 

4785 

4916 

335 

6045 

6174 

6304 

5434 

5563 

6693 

6822 

5951 

6081 

6210 

336 

’ 6339 

6469 

6598 

6727 

6856 

6985 

7114 

7243 

7372 

7501 

337 

7630 

7759 

7888 

8016 

8145 

8274 

8402 

8631 

8660 

8788 

338 

8917 

9045 

9174 

9302 

9430 

9559 

9687 

9815 

9943 

..72 

339 

630200 

0328 

0456 

0584 

0712 

0840 

0968 

1096 

1223 

1361 

340 

1479 

1607 

1734 

1862 

1960 

2117 

2246 

2372 

2600 

2627 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3618 

3645 

3772 

3899 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

6167 

343 

6294 

6421 

6547 

5674 

5800 

6927 

6053 

6180 

6306 

6432 

344 

6558 

6685 

6811 

6937 

7063 

126 

7189 

7315 

7441 

7667 

7693 

345 

7819 

7946 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9964 

. .79 

.204 

347 

540329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1464 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2676 

2701 

349 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 
















































8 

• 


LOGARITHMS 




N. 

0 

1 

2 

3 

4 

5 ' 

6 

7 

8 

9 

350 

644068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

6060 

5183 

351 

5307 

5431 

5555 

5678 

6805 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8768 

8881 

354 

9003 

9126 

9249 

9371 

9494 

122 

9616 

9739 

9861 

9984 

.196 

355 

650228 

0351 

0473 

0696 

0717 

0840 

0962 

1084 

1206 

1328 

350 

1460 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3393 

3519 

3640 

3762 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

359 

6094 

6215 

5346 

6457 

6578 

6699 

6820 

6940 

6061 

6182 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

301 

7607 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8689 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

363 

9907 

. .26 

.146 

.265 

.386 

.604 

.624 

.743 

.863 

.982 

364 

561101 

1221 

1340 

1459 

1678 

1698 

1817 

1936 

2056 

2173 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

366 

3481 

3600 

3718 

3837 

8965 

4074 

4192 

4311 

4429 

4548 

367 

4666 

4784 

4903 

5021 

5139 

5257 

6376 

6494 

6612 

5730 

368 

6848 

6966 

6084 

6202 

6320 

6437 

6565 

6673 

6791 

6909 

369 

7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

370 

8202 

8319 

8436 

8554 

8671 

8788 

8905 

9023 

9140 

9267 

371 

9374 

9491 

9608 

9726 

9882 

9959 

. .76 

.193 

.309 

.426 

372 

670543 

0660 

0776 

0893 

1010 

1126 

1243 

1369 

1476 

1592 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2623 

2639 

2765 

374 

2872 

2988 

3104 

3220 

3336 

116 

3452 

3668 

3684 

3800 

1915 

375 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

376 

6188 

5303 

5*19 

5534 

5650 

5766 

5880 

6996 

6111 

6226 

377 

63 il 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9665 

9669 

380 

9784 

9898 

..12 

.126 

.241 

.355 

.469 

.583 

.697 

.811 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2868 

2972 

3085 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5236 

5348 

385 

5461 

6574 

5686 

6799 

5912 

6024 

6137 

6250 

6362 

• 6475 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7699 

387 

7711 

7823 

7936 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

388 

8832 

8944 

9056 

9167 

9279 

9391 

9603 

9615 

9726 

9834 

389 

9950 

. .61 

.173 

.284 

.396 

.507 

.619 

.730 

,S42 

.963 

390 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2. 66 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

6276 

5386 

394 

6496 

5606 

6717 

5827 

5937 

110 

6047 

6157 

6267 

6377 

6487 

395 

6597 

6707 

6817 

6927 

7037 

<7146 

7256 

7366 

7476 

7586 

396 

7696 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8672 

8681 

397 

8791 

8900 

9009 

9119 

9228 

9337 

9446 

1656 

9666 

9774 

398 

9883 

9992 

.101 

.210 

.319 

.428 

.537 

.646 

765 

.864 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 
-- i 

1951 



































OF NUMBERS. 9 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400 

602030 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

401 

3144 

3253 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

402 

4226 

4334 

4442 

4650 

4658 

4766 

4874 

4982 

5089 

6197 

403 

5305 

5413 

5521 

5628 

5736 

5844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

108 

6919 

7026 

7133 

7241 

7348 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

407 

9594 

9701 

9808 

9914 

. .21 

.128 

.234 

.341 

.447 

.554 

403 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

412 

4897 

6003 

5108 

6213 

5319 

5424 

5529 

6634 

5740 

5845 

413 

6950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

7943 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

416 

9293 

9198 

9302 

9406 

9511 

9615 

9719 

9824 

9928 

. .32 

417 

620136 

0140 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

418 

1176 

1280 

1384 

1488 

1692 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

6004 

5107 

6210 

422 

5312 

5415 

6518 

5621 

5724 

5827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

6761 

6853 

6956 

7058 

7161 

7263 

424 

7366 

7468 

7571 

7673 

7776 

103 

7878 

7980 

8082 

8185 

8287 

425 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

S206 

9308 

426 

9410 

9512 

9613 

9716 

9817 

9919 

. .21 

.123 

.224 

.326 

427 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139 

1241 

1342 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

431 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

6383 

432 

5484 

5584 

5685 

5785 

5886 

6986 

6087 

6187 

6287 

6388 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

434 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

8290 

8389 

435 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

436 

9486 

9586 

9686 

9785 

9885 

9984 

. .84 

.183 

.283 

.382 ' 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 i 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

| 440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

1 441 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226 

6324 

442 

5422 

5521 

6619 

5717 

5815 

6913 

6011 

6110 

6208 

6306 

443 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

444 

7383 

7481 

7679 

7676 

7774 

98 

8750 

7872 

7969 

8067 

8165 

8262 

445 

8360 

8458 

8555 

8653 

8848 

8945 

9043 

S140 

9237 

446 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

. .16 

.113 

.210 

447 

650308 

0405 

0502 

0599 

0696 

0/93 

0890 

0987 

1084 

1181 

418 

1278 

13/5 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

449 

2246 

2343 

2440 

2530 

2633 

2730 

2826 

2923 

3019 

3116 










































10 LOGARITHMS 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

450 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

451 

4177 

4273 

4359 

4466 

4562 

4658 

4764 

4850 

4946 

6042 

452 

6138 

5235 

5331 

5427 

5526 

5619 

5715 

5810 

6906 

6002 

453 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

454 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

455 

8011 

8107 

8202 

8298 

JO 

8393 

8488 

8584 

8679 

8774 

8870 

456 

8966 

9u60 

9155 

9250 

9346 

9441 

<636 

9631 

9726 

9821 

457 

9916 

1 i 

.106 

.201 

.296 

.391 

.486 

.581 

.676 

.771 

458 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1629 

1623 

1718 

459 

1813 

190/ 

2002 

2006 

2191 

2286 

2380 

2475 

2669 

2663 

460 

2758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3612 

8607 

461 

3701 

3795 

388S 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

462 

4642 

4736 

48 0 

4924 

5018 

5112 

6206 

5299 

5393 

6487 

463 

/ 6581 

5676 

6759 

6862 

6956 

6050 

6143 

6237 

6331 

6424 

464 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

465 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

466 

8386 

8479 

8672 

8665 

8759 

8852 

8945 

9038 

9131 

9324 

467 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

. .60 

.153 

468 

670241 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

469 

1173 

1265 

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1 

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OF NUMBERS. 



11 

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3353 

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0 

F NUMBERS. 



13 

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4 

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6 

7 

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14 


LOGARITHMS 


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4447 

4514 

4581 

4647 

4714 

4/80 

4847 

653 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

6378 

6445 

5511 

654 

55 78 

5644 

5/11 

5777 

5843 

67 

5910 

59/6 

6042 

6109 

6175 

655 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

656 

6904 

6970 

7036 

7102 

7169 

7233 

7301 

7367 

7433 

7499 

657 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

658 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

659 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

...4 

..70 

.136 

661 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

0661 

0727 

0792 

662 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

663 

1514 

15/9 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

664 

2168 

2233 

2299 

2364 

2430 

2496 

2560 

2626 

2691 

2766 

665 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

666 

3474 

3539 

3605 

3670 

3735 

3800 

3866 

3930 

3996 

4061 

667 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

47 ; 46 

4711 

668 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

669 

6426 

6491 

5556 

5621 

5686 

5751 

5815 

5880 

6945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

672 

7369 

7434 

749f 

7563 

7628 

7692 

7767 

7821 

7886 

7951 

fw.'t 

8015 

8080 

814 * 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

675 

9304 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

676 

9947 

. .11 

. .75 

.139 

.204 

.268 

.332 

.396 

.460 

.525 

677 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

678 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

16,8 

1742 

1806 

679 

1870 

1934 

, 1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

680 

2509 

2573 

! 2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

681 

3147 

3211 

1 3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

682 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

683 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

684 

6056 

5120 

5183 

5247 

6310 

5373 

5437 

5500 

5564 

5627 

685 

6691 

5754 

' 6817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

686 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

. 687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7526 

688 

7688 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

689 

8219 

8282 

8345 

| 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

690 

8849 

8912 

8975 

9038 

9109 

9164 

9227 

9289 

9352 

9415 

691 

9478 

9541 

9604 

9667 

9729 

9792 

9856 

9918 

9981 

. .43 

692 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

693 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

694 

1359 

1422 

1485 

1547 

1610 

62 

1672 

1735 

1797 

1860 

1922 

695 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2647 

696 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

697 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

698 

3865 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

699 

4477 

4539 

4601 

4664 

i 

4726 

4788 

4850 

4912 

4974 

5036 
































































OF NUMBERS. 1‘5 


N. 

0 

1 

2 

3 

f 

4 

5 

6 

7 

8 

9 

700 

845098 

5160 

5222 

5284 

5346 

6408 

5470 

6532 

6594 

5656 

701 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6276 

702 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

703 

6955 

7017 

70/9 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

704 

7673 

7634 

7676 

7758 

7819 

62 

7881 

7943 

8004 

8066 

8128 

705 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

700 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

707 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

708 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

709 

0648 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

710 

1258 

1320 

1381 

1442 

1503 

1664 

1625 

1686 

1747 

1809 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

713 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

714 

3698 

3759 

3820 

3881 

3941. 

4002 

4063 

4124 

4185 

4245 

715 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

716 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

6398 

6459 

717 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

718 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

719 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

720 

7332 

7393 

7453 

7613 

7574 

7634 

7694 

7755 

7816 

7875 

721 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

723 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

95 j 9 

9619 

9679 

724 

9739 

9799 

9859 

9918 

9978 

65 

..38 

..98 

.158 

.218 

.278 

725 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

726 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

727 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

728 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

729 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

731 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

732 

4511 

4570 

4630 

4689 

4148 

4808 

4867 

4926 

4985 

6045 

733 

6104 

6163 

5222 

6282 

6341 

6400 

5459 

6519 

6578 

6637 

734 

6696 

5755 

5814 

6874 

5933 

6992 

6051 

6110 

6169 

6228 

735 

6287 

6346 

6405 

6465 

6624 

6583 

6642 

6701 

6760 

6819 

736 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

737 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

738 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

739 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

9173 

740 

9232 

9290 

9349 

9408 

9466 

9525 

9584 

9642 

9701 

9760 

741 

9818 

9877 

9935 

9994 

. .63 

.111 

.170 

.228 

.287 

.345 

742 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

743 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1615 

744 

1573 

1631 

1690 

1748 

1806 

59 

1865 

1923 

1981 

2040 

2098 

745 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

746 

2739 

2797 

2855 

2913' 

2972 

3030 

3088 

3146 

3204 

3262 

747 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3786 

3844 

748 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4360 

4424 

749 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4946 

5003 
























































16 



LOGARITHMS 




N. 

0 

l 

2 

3 

4 

5 

6 

7 

8 

9 

750 

875061 

- 1 

6119 

5177 

6235 

5293 

5351 

5409 

5466 

5524 

5582 

751 

6610 

5693 

5756 

5813 

6871 

5929 

5987 

6045 

6102 

6160 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

754 

7371 

7429 

7487 

7544 

7602 

57 

8177 

7659 

7717 

7774 

7832 

7889 

755 

7947 

8004 

8062 

8119 

8234 

8292 

8349 

8407 

8464 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8951 

9039 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

758 

9669 

9726 

9784 

9841 

9898 

9956 

. .13 

. .70 

.127 

.185 

759 

830242 

0299 

0356 

0413 

0471 

0628 

0580 

0542 

0699 

0756 

760 

0314 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

761 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

762 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

763 

2525 

2581 

2638 

2695 

2762 

2809 

2866 

2923 

2980 

3037 

764 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

765 

3661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

766 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

767 

4 795 

4852 

4909 

4965 

6022 

5078 

5135 

5192 

5248 

5305 

768 

5361 

5418 

5474 

5531 

6587 

5644 

5700 

5757 

6813 

5870 

769 

6926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

771 

7054 

7111 

7167 

7233 

7280 

7336 

7392 

7449 

7505 

7561 

772 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

773 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8655 

774 

8741 

8797 

8853 

8909 

8965 

56 

9021 

9077 

9134 

9190 

9246 

775 

9302 

9358 

9414 

9470 

9526 

9582 

9638 

9694 

9760 

9806 

776 

9862 

9918 

0974 

. .30 

. .86 

.141 

.197 

.253 

.309 

.365 

777 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

778 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

1370 

1426 

1482 

779 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

781 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3010 

3096 

3151 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

784 

4316 

4371 

4427 

4482 

4538 

4693 

4648 

4704 

4759 

4814 

785 

4870 

4925 

4980 

6036 

6091 

6146 

5201 

6257 

6312 

5367 

786 

6423 

5478 

6533 

6588 

6644 

6699 

6754 

6809 

6864 

5920 

787 

6975 


6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

. 788 

6626 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

789 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

790 

7627 

7683 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

| 791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

792 

8726 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

793 

9273 

9328 

9383 

9437 

9492 

9547 

9602 

9666 

9711 

9766 

794 

9821 

9875 

9930 

9985 

..39 

55 

. .94 

.149 

.203 

.258 

.312 

795 

900367 

0422 

0476 

0531 

0586 

0640 

0695 

0749 

0804 

0859 

796 

0913 

0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

79 7 

1458 

1513 

1567 

1622 

1676 

1736 

1785 

1840 

1894 

1948 

798 

2003 

2067 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2<‘81 

3036 

- n 
















































OF NUMBERS. 


17 


N. 

0 

1 

•2 

3 

4 

6 

6 

7 

8 

It 

80;) 

903000 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

801 

3633 

3387 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

803 

4716 

4770 

4824 

4878 

4932 

4983 

5040 

5094 

5148 

5202 

804 

6256 

5310 

5364 

5418 

6472 

54 

6526 

6580 

5634 

5088 

5742 

F05 

6796 

5850 

5904 

5958 

6012 

6036 

6119 

6173 

6227 

6281 

803 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

807 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

808 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

800 

7949 

8002 

8058 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

810 

8485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

811 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

812 

9558 

9810 

9563 

9716 

9770 

9823 

9877 

9930 

9984 

. .37 

813 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0571 

814 

0824 

06 78 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

810 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2115 

2169 

817 

2222 

2275 

2323 

2381 

2435 

2488 

2541 

2594 

2645 

2700 

818 

2753 

2806 

2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

819 

3284 

3337 

3390 

3443 

3496 

3549 

3602 

3655 

3708 

3761 

820 

3814 

3887 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

821 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

822 

4872 

4925 

4977 

5030 

6083 

5136 

5189 

5241 

5294 

5347 

823 

5400 

5453 

6505 

5558 

5611 

5664 

5716 

6769 

5822 

5875 

824 

6927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

825 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

826 

6980 

7033 

7035 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

827 

7503 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

828 

8030 

8033 

8185 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

829 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

831 

9601 

9653 

9705 

9758 

9810 

9862 

9914 

9967 

. .19 

. .71 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

833 

0845 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

834 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

836 

2206 

2258 

2310 

2362 

2414 

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3451 

3503 

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3607 

3658 

3710 

839 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4147 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

841 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

6261 

842 

6312 

5364 

5415 

5467 

5518 

5570 

5621 

6673 

5725 

5776 

843 

5828 

5874 

5931 

5982 

6034 

6035 

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6188 

6240 

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844 

6342 

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6445 

6497 

6548 

6600 

6651 

6702 

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6805 

845 

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6908 

6959 

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7114 

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7216 

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7319 

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7370 

7422 

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7730 

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847 

7883 

7935 

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8037 

8088 

8140 

8191 

8242 

8293 

8345 

848 

8395 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

849 

8903 

8959 

9010 

9031 

9112 

9163 

9216 

9266 

9317 

9368 











— 













































18 L O G A R I T H J1 S 


N. 

0 

l 1 

1 

0 

3 

4 

6 

6 

7 

8 

9 

850 

929419 

9473 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

851 

9930 

9981 I 

. .32 

. .83 

.134 

.185 

.236 

.287 

.338 

.389 

852 

930440 

0491 1 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

853 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

854 

1458 

1509 

1560 

1610 

1661 

51 

1712 

1763 

1814 

1865 

1915 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

80S 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

859 

3993 

4044 

4094 

4145 

4195 

4246 

4269 

4347 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

861 

5093 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

862 

550 7 

5558 

6608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

863 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

864 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

865 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

867 

8019 

8069 

8119 

8169 

8219 

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8320 

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8420 

8470 

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8720 

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869 

9020 

9070 

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9170 

9220 

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9320 

9369 

9419 

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870 

9519 

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9816 

9669 

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9769 

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9869 

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9968 

871 

940018 

0068 

0118 

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0317 

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0417 

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0566 

0616 

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0716 

0765 

0815 

0865 

0915 

0964 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

876 

2504 

2554 

2603 

26.3 

2702 

2752 

2801 

2851 

2901 

2950 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

878 

3495 

3544 

3693 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4632 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5025 

6074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

882 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

883 

6961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

885, 

6043 

6992 

7041 

7090 

7146 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

i365 

888 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

889 

8902 

8951 

8999 

9018 

9097 

9146 

9195 

9244 

9292 

4341 

890 

9390 

9439 

9488 

9536 

9586 

9634 

9683 

9731 

9780 

9829 

891 

9878 

9926 

9975 

..24 

. .73 

.121 

. 1/0 

.219 

.267 

.316 

892 

960365 

0414 

0162 

0511 

0560 

0608 

0657 

0/06 

0754 

080? 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

894 

1338 

1386 

1435 

1483 

1532 

1580 

162J 

1677 

1726 

1775 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

896 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

5696 

^744 

897 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

898 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

899 

3760 

3808 

3856 

3905 

3953 

4001 

4019 

40J8 

4146 

4194 

















































r 


OF NUMBERS. ID 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5168 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

6543 

5592 

6640 

903 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6605 

6663 

6601 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7659 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9011 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

912 

9995 

. .42 

. .90 

.138 

.185 

.233 

.280 

.328 

.376 

.423 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

919 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

6061 

5108 

5155 

923 

5202 

5249 

6296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

924 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7464 

7501 

928 

7548 

7695 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8866 

8903 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9976 

. .21 

. .68 

.114 

.161 

.207 

.254 

.300 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1926 

1971 

2018 

2064 

2110 

2157 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

941 

3590 

3836 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4*120 

4466 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

944 

4972 

6018 

6064 

5110 

5156 

46 

5202 

5248 

5294 

5340 

5386 

945 

5432 

5478 

5524 

5570 

6616 

5662 

6707 

5753 

5799 

5845 

946 

6891 

5937 

6983 

6029 

6075 

6121 

6167 

6212 

6268 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

<1671 

6717 

6763 

948 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

949 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

767« 








































20- LOGARITHMS 


N. 

0 

l 

2 

3 

4 

6 

6 

7 

8 

9 

950 

977724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

951 

8181 

8226 

8272 

8317 

,'■363 

8409 

8454 

8500 

8546 

8591 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

954 

9548 

9594 

9639 

9685 

9730 

46 

9776 

9821 

9867 

9912 

9958 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

95o 

0458 

0503 

0549 

0594 

0340 

0685 

0730 

0776 

0821 

0867 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

962 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

964 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5067 

6112 

5157 

5202 

5247 

6292 

5337 

5382 

967 

5426 

6471 

5516 

5661 

5606 

5651 

6699 

5741 

5786 

6830 

968 

5875 

6920 

6965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8426 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

975 

9005 

9049 

9093 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

977 

9895 

9939 

9983 

. .28 

. .72 

.117 

.161 

.208 

.250 

.294 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1936 

1979 

2u23 

2067 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2466 

2609 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4886 

4933 

4977 

5021 

5065 

6108 

5152 

989 

5196 

5240 

5284 

5328 

6372 

5416 

6460 

6504 

6547 

6591 

990 

5635 

6679 

5723 

6767 

6811 

6854 

5898 

5942 

5986 

6030 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6612 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

1 993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7617 

7661 

44 

7605 

7648 

7692 

7736 

7779 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8662 

997 

8695 

8739 

8792 

8826 

8869 

8913 

8966 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

999 

. -- 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

i 





































|-- - ~~ -^vIWwhi ■ ii m i i, ^ 



TABLK Ii 

! 

-«;j. Sines 

and Tangents. 

(0°) Natural Sines 

21 

t 

Sine. 

D 10' 

'| Cos: lie. 

D. 10' 

Tang. 

D. 10' 

Coiaug. 

N.sine 

. N. cos. 


0 

1 

2 

3 

4 
6 
6 

7 

8 

9 

10 
il 
<2 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 
29 
39 
31 
52 
•33 

34 

35 
38 

37 

38 

39 

40 

41 
49 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 
65 

56 

57 

58 

59 

60 

Minn.-- rj 
6.463'26 
7S475> 
910847 
7 .055786 
162693 
241877 
308824 
368816 
417968 
483725 
7.605118 
542903 
677668 
609853 
639816 
667845 
694173 
718997 
742477 
764754 
7.785943 
806143 
825451 
843934 
861663 
?78695 
895085 
910879 
926119 
940842 
7.955032 
968870 
982233 
995198 
8.007787 
020021 
031919 
043501 
054781 
035776 
8.076500 
086965 
097183 
107167 
116926 
126471 
135810 
144953 
153907 
162681 

3.171280 
179713 
187985 
196102 
204070 
211895 
219581 
227134 
234557 
241855 

1 

•1 

2298 

2227 

2161 

2098 

2039 

1983 

1930 

1880 

1832 

1787 

1744 

1703 

1664 

1626 

1591 

1657 

1524 

1492 

1462 

1433 

1405 

1379 

1353 

1328 

1304 

1281 

1259 

1237 

1216 

10.00000C 

000000 

000000 

000090 

000900 

000000 

9.999999 

999999 

999999 

999999 

999998 

9.999998 

999997 

999997 

999996 

999996 

999995 

999995 

999994 

999993 

999993 

9.999992 

999991 

999990 

999989 

999988 

999988 

999987 

999986 

999985 

999983 

9.999982 

999981 

999980 

999979 

999977 

999976 

999975 

999973 

999972 

999971 

9.999969 

999968 

999966 

999964 

999963 

999961 

999959 

999958 

999956 

999954 

9.999952 

999950 

999948 

999946 

999944 

999942 

999940 

999938 

999936 

999934 

0.2 

0.2 

0-2 

0-2 

0-2 

0-2 

0-2 

0-2 

02 

02 

02 

0‘2 

0 2 

0 3 

0 3 

0 3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.4 
0.4 
0.4 
0.4 

Minus inf 
6.463726 
764756 
940847 
7.065786 
162696 
241878 
308825 
366817 
417970 
463727 
7.505120 
542909 
677672 
609857 
639820 
667849 
694179 
719003 
742484 
764761 
7.785951 
806155 
825460 
843944 
861674 
878708 
895099 
910894 
926134 
940858 
7.955100 
968889 
982263 
995219 
8.007809 
020045 
031945 
043527 
054809 
065806 
8.076531 
086997 
097217 
107202 
116963 
126610 
135851 
144996 
163952 
162727 
3.171328 
179763 
188036 
196156! 
204126! 
211953i 
219641 
227195 
234621 
241921 

2298 

2227 

2161 

2098 

2039 

1983 

1930 

1880 

1833 

1787 

1744 

1703 

1664 

1627 

1591 

1557 

1524 

1493 

1463 

1434 

1406 

1379 

1353 

1328 

1304 

1281 

1269 

1238 

1217 

Infinite. 
13.536274 
235244 
059153 
12.934214 
837304 
758122 
691175 
633183 
582030 
636273 
12.494880 
457091 
422328 
390143 
360180 
332151 
305821 
280997 
257516 
235239 
12.214049 
193845 
174540 
156056 
138326 
121292 
104901| 
089106) 
073866) 
059142 
12.044900 
031111 
017747 
004781 
11.992191 
979955 
968056 | 
956473 
945191 | 
934194 
11.923469) 
913003!) 
902783)1 
892797 : 
883037 | 
873490)! 
864149 
855004 ) 
846048 | 
837273 

11.828672 
820237 
811964 
803844 
795874 
788047 
780359 
772805 
765379 
753079 

0000! 

00021 

00051 

0003' 

00111 

00147 

00177 

0020! 

00237 

00265 

00291 

0032! 

00347 

00378 

00407 

00436 

00465 

00495 

00524 

00553 

00582 

00611 

00640 

00669 

00698 

00727 

00756 

00786 

00814 

00844 

00873 

00902 

00931 

00960 

00989 

01018 

01047 

01076 

01105 

01134 

01164 

01193 

01222 

01251 

01280 

01309 

01338 

01367 

01396 

01425 

01454 

01483 

01513 

01542 

01571 

01600 

01629 

01668 

01687 

01716 

01745 

)100001 
) 100001 
; looooi 
100001 
100001 
100001 
10000! 
icoooc 
100001 
10000C 
,1000X1 
>! 99999 
>j 99999 
1 99999 
99999 
99999 
99999 
99999 
99999 
99998 
99998 
99998 
99998 
99998 
99998 
99997 
99997 
99997 
99997 
99996 

99993 
99996 
93996 
99995 
99995 
99995 
99995 

99994 
99994 
99994 
99993 
99993 
99993 
99992 
99992 
99991 
99991 
99991 
99990 
99990 
99989 
99989 
99989 
99988 
99988 
99987 
99987 
99986 
99986 
99985 
99985 

) 60 
59 
58 
57 
56 
55 
64 
53 
62 
51 
50 
49 
48 

47 

48 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

15 

17 

16 
]5 

14 

13 

12 

11 

10 i l - 

9 ! 
8 

7 

6 

5 

4 i 
3 1 
2 

1 i 
0 


Cosine. 


S:ne. 


Civansr. 


'ran!? 

N. cos. 

V. sine - 

/ 


S9 Degrees. 
















































































































22 Log. Sines and Tangents. (1°) Natural Sines. TABLE II. 


i 

Sine. 

D 10"| 

Cosine. 

0.10" 

Tang. 

D lU" 

Coiang. 

N. sine. 

Jf. cos. 


0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
66 

57 

58 

59 

60 

5.241855 
249033 
256094 
263042 
269881 
276 514 
283243 
289773 
296207 
302545 
308794 
8.314954 
321027 
327016 
332924 
338753 
344504 
350181 
355783 
361315 
366777 
8.372171 
377499 
382762 
387962 
393101 
398179 
403199 
408161 
413068 
417919 
8.422717 
427462 
432156 
436800 
441394 
445941 
450440 
454893 
459301 
463665 
8.467985 
472263 
476498 
480693 
484848 
488963 
493040 
497078 
501080 
505045 
8.508974 
512867 
516726 
520551 
524343 
528102 
531828 
535523 
539186 
542819 

-1 

u 

1196 
1177 
1158 
1140 
1122 
1105 
1088 
1072 
1056 
1041 
1027 
1012 
998 
985 
971 
959 
946 
934 
922 
910 
899 
888 
877 
867 
856 
846 
837 
827 
818 
809 
800 
791 
782 
774 
766 
758 
! 750 
742 
735 
727 
720 
712 
706 
699 
692 
686 
679 
673 
667 
661 
655 
649 
643 
637 
632 
626 
621 
616 
611 
605 

999934 
999932 
999929 
999927 
999925 
999922 
999920 
999918 
999915 
999913 
999910 
9.999907 
999905 
999902 
999899 
999897 
999894 
999891 
999888 
999885 
999882 
9.999879 
999876 
999873 
999870 
999867 
999864 
999861 
999858 
999854 
999851 
9.999848 
999844 
999841 
999838 
999834 
999831 
999827 
999823 
999820 
999816 
9.999812 
999809 
999805 
999801 
999797 
999793 
999790 
999786 
999782 
999778 
9.99977- 
999769 
999765 
999761 
99975 7 
999753 
999748 
999744 
999740 
999735 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0.4 

0-4 

0-4 

0-4 

04 

0.4 

0.5 

0.5 

0-5 

05 

0-5 

0.5 

0.5 

05 

0-5 

0.5 

0.5 

06 

0.5 

0.5 

0.5 

0-5 

0-6 

0-6 

0.6 

0-6 

0-6 

0.6 

0.6 

0.6 

0-6 

0.6 

06 

0-6 

0-6 

0.6 

0.6 

0-7 

0-7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

3.241921 

249102 

256165 

263115 

269956 

276691 

283323 

289856 

296292 

302634 

308884 

8.315046! 

321122 

327114 

333025 

333856 

344610 

350289 

355895 

361430 

366895 

8.372292 

377622 

382889 

388092 

393234 

398315 

403338 

408304 

413213 

418068 

8.422869 

427618 

432315 

436962 

441560 

446110 

450613 

455070 

-459481 

463849 

8.468172 

472454 

476693 

480892 

485050 

489170 

493250 

497293 

501298 

505267 

8.509200 

513098 

516961 

520790 

524586 

528349 

532080 

535779 

539447 

543084 

1197 

1177 

1158 

1140 

1122 

1105 

1089 

1073 

1057 

1042 

1027 

1013 

999 

985 

972 

959 

946 

934 

922 

911 

899 

888 

879 

867 

857 

847 

837 

828 

818 

809 

800 

791 

783 

774 

766 

758 

750 

743 

735 

728 

720 

713 

707 

700 

693 

686 

680 

674 

668 

661 

655 

650 

644 

638 

633 

627 

622 

616 

611 

606 

11.758079 
750898 
743835 
736885 
730044 
723309 
716677 
710144 
703708 
697366 
691116 
11-684954 
678878 
672886 
666975 
661144 
655390 
649711 
644105 
638570 
633105 
11-627708 
622378 
617111 
611908 
606766 
601685 
596662 
591696 
586787 
581932 
11.577131 
572382 
567685 
563038 
658440 
653890 
549387 
544930 
540519 
636151 
11-531828 
527546 
623307 
519108 
514950 
610830 
606750 
602707 
498702 
49-4733 
11.490800 
486902 
483039 
479210 
475414 
471651 
467920 
464221 
460553 
456916 

01742 
01774 
01803 
01832 
01862 
01891 
01920 
01949 
01978 
02007 
021136 
02065 
02094 
02123 
02152 
02181 
02211 
02240 
02269 
02298 
02327 
02356 
02385 
02414 
02443 
02472 
02501 
02530 
02560 
i02589 
02618 
02647 
|02676 
i02705 
02734 
02763 
02792 
02821 
02850 
102879 
i02908 
02938 
02967 
02991) 
03025 
03054 
03083 
103112 
103141 

103170 
! 03199 
103228 
!03257 
03286 
03316 
03345 
03374 
03403 
03432 
03461 
03490 

49985 

49984 

99984 

99983! 

99983 1 

999821 

99982 

99281 

99980 

99980 

99979 

99979 

99978 

99977 

96977 

99976 

99976 

99975 

99974 

91)974 

99973 

9»972 

99972 

99971 

99970 

69969 

99969 

99968 

99967 

99966 

99966 

99965 

99964 

99963 

99963 

99962 

99961 

99960 

99959 

99959 

99958 

99957 

99956 

99955 

99954 

99953 

99952 

99952 

99951 

99950 

99949 

99948 

99947 

99946 

9d945 

99944 

99943 

99942 

99941 

9994i 

99939 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
! 12 
11 
10 
9 
8 
7 
6 
6 
4 
3 
2 
1 
0 


Cos’ne. 

Sine. 


Cotang. 


Tang. 

N. cos 

N -sine 

t 


88 Degree*. 



































































TABLK II 

I*°g. 8;ncs and Tangents. (2°) Natural Sines. 


23 


> n*- 

L>. l»"| Cosine. 

D 10' 

Tang. 

D. 10"i Cotang. 

! |N. sine 

. X. eos 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

* 10 
11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 
! 40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 
65 

56 

57 
68 

; 59 
60 

8.542819 
546422 
549995 
553539 
557054 
560540 
563999 
667431 
570836 
574214 
577566 
8.580892 
584193 
587469 
590721 
593948 
697152 
. 600332 
603489 
606623 
609734 
8.612823 
615891 
618937 
621962 
624965 
627948 
630911 
633854 
636776 
639680 
8.642563 
645428 
648274 
651102 
653911 
656702 
659475 
662230 
664968 
667689 
8.670393 
673080 
675751 
678405 
681043 
683665 
686272 
688863 
691438 
693998 

3.696543 
699073 
701589 
704090 
706577 
709049 
711507 
713952 
716383 
718800 

60) 

595 

591 

586 

681 

576 

572 

567 

663 

659 

554 

550 

546 

542 

538 

534 

530 

526 

522 

519 

515 

611 

508 

504 

601 

497 

494 

490 

487 

484 

481 

477 

474 

471 

468 

465 

462 

459 

456 

453 

451 

448 

445 

442 

440 

437 

434 

432 

429 

427 

424 

422 

419 

417 

414 

412 

410 

407 

405 

403 

9.999735 

999731 

999726 

999722 

999717 

999713 

999708 

999704 

999699 

999694 

999689 

9.999685 

999680 

999676 

999670 

999665 

999660 

999655 

999650 

999645 

999640 

9.999635 

999629 

999324 

999619 

999614 

999608 

999603 

999597 

999592 

999586 

9.999581 

999576 

999570 

999564 

999558 

999553 

999547 

999541 

999535 

999629 

9.999524 

999518 

999512 

999506 

999500 

999493 

999487 

999481 

999475 

999469 

9.999463 

939456 

999450 

999443 

999437 

999431 

999424 

999418 

999411 

999404 

0.7 

0.7 

0-7 

0-8 

0-8 

0-8 

0.8 

0.8 

0-8 

0-8 

0.8 

0-8 

0-8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0-9 

O.y 

0-9 

09 

0-9 

0.9 

0.9 

0-9 

0.9 

0.9 

0.9 

0.9 

1.0 

10 

1.0 

1.0 

10 

1.0 

1.0 

1.0 

10 

1-0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.0 

1.1 

1 1 
1.1 
1.1 

1 1 
1.1 
1.1 
1.1 
1.1 

8.543984 
546691 
550268 
553817 
557336 
660828 
564291 
567727 
671137 
574520 
577877 
8.581208 
584514 
687795 
691051 
594283 
597492 
600677 
603839 
606978 
610094 
8.613189 
616262 
619313 
622343 
625352 
628340 
631308 
634256 
637184 
640093 
8.642982 
645853 
648704 
651537 
654352 
657149 
659928 
662689 
665433 
668160 
8.670870 
673563 
676239 
678900 
681644 
684172 
6 6784 
689381 
691963 
694529 
B. 697081 
699617 
702139 
704246 
707140 
709618 
702083 
714534 
716972 
719396 

602 

596 

591 

587 

582 

577 

573 

668 

564 

559 

555 

651 

647 

543 

539 

535 

531 

627 

623 

519 

516 

512 

508 

505 

601 

498 

495 

491 

488 

485 

482 

478 

475 

472 

469 

466 

463 

460 

457 

454 

453 

449 

446 

443 

442 

438 

435 

433 

430 

428 

425 

423 

420 

418 

415 

413 

411 

408 

406 

404 

11.456916 
453309 
449732 
446183 
442664 
439172 
435709 
432273 
428863 
425480 
422123 
11.418792 
415486 
412205 
408949 
405717 
402508 
399323 
396161 
393022 
389906 
11.386811 
383738 
380687 
377657 
374648 
371660 
368692 
366744 
362816 
359907 
11.357018 
354147 
351296 
348463 
345648 
342851 
340072 
337311 
334567 
331840 
11.329130 
326437 
323761 
321100 ! 
318456 r 
315828 1 
313216 
310819 
308037 
305471 
11.302919 
300383i 
297861 
295354 
292860 
290382 
287917 
285465 
283028 ! 
280604 

03491 
0351 b 
03548 
03577 
03606 
03635 
103664 
: 03693 
!03723 
03752 
03781 
03810 
03839 
03868 
03897 
03926 
03955 
03984 
04013 
04042 
04071 
04100 
03129 
04159 
04188 
04217 
04246 
04275 
04304 
04333 
04362 
04391 
04420 
04449 
04478 
04507 
04536 
04565 
04594 
04623 
04653 
04682 
04711 
04740 
04769 
04798 
04827 
04856 
04885 
04914 
04943 
04972 
05001 
05030 
05059 
05088 
05117 
05146 
05175 
05205 
05234 

99931 

9993£ 

94931 

99931 
99935 
99934 
99933 

99932 
99931 
99930 
99929 
99927 
99926 
99925 
99924 
99923 
99922 
99921 
99919 
9991c 
99917 
99916 
99915 
99913 
99912 
99911 
99910 
99909 
99907 
99906 
99905 
99904 
99902 
99901 
99900 
99898 
99897 
99896 
99894 
49893 
99892 
99890 
99889 
99888 
99886 
49885 
99883 
99882 
99881 
99879 
99878 
99876 
99875 
99873 
99872 
99870 
99869 
9i>867 
99866 
19864 
99863 

0 60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 1 
47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

33 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

1 

Cosine. 


Sine. 


Cotang. 


Tang. J 

N. cos. 

N.sine. 

1 





87 Degrees. 








































































24 Log. Sines and Tangents. {-P) Natural Sines. TABLE II. 


/ 

Sine. 

0. id' 

Cosine. 

D. 1 i 

Tang. 

9. Id" 

Co tang. ||N.sine 

N. cos 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 
62 
63 

54 

55 

56 

57 

58 

59 

60 

8.718800 
721204 
723595 
725972 
728337 
730688 
733027 
735354 
737667 
739969 
742259 
8.744536 
746802 
749055 
751297 
753528 
755747 
757955 
760151 
762337 
764511 
8.766675 
768828 
770970 
773101 
775223 
7773S3 
779434 
781524 
783605 
785675 
8.787736 
789787 
791828 
793869 
795881 
797894 
799897 
801892 
803876 
805852 
8.807819 
809777 
811726 
813667 
815599 
817522 
819436 
821343 
823240 
825130 
8.827011 
828884 
830/49 
832607 
834456 
836297 
838130 
839956 
- 841774 
843585 

401 

398 

396 

394 

392 

390 

388 

386 

384 

382 

380 

378 

376 

374 

372 

370 

368 

366 

364 

362 

361 

359 

357 

355 

353 

352 

350 

348 

347 

345 

343 

342 

340 

339 

337 

335 

334 

332 

331 

329 

328 

326 

325 

323 

322 

320 

319 

318 

318 

315 

313 

312 

311 

309 

308 

307 

308 

301 
303 

302 

9.999404 

999398 

999391 

999384 

999378 

999371 

999364 

999357 

999350 

999343 

999336 

9.999329 

999322 

999315 

999308 

999301 

999294 

999286 

999279 

999272 

999265 

9.999257 

999250 

999242 

999235 

999227 

999220 

999212 

999205 

999197 

999189 

9.999181 

999174 

999166 

999158 

999150 

999142 

999134 

999126 

999118 

999110 

9.999102 

999094 

999086 

999077 

999069 

999061 

999053 

999044 

999036 

999027 

9.999019 

999010 

999002 

998993 

998984 

998976 

998967 

998958 

998950 

998941 

1.1 

1.1 

1.1 

1.1 

1.1 

1.1 

1.2 

1.2 

1.2 

1.2 

1.2 

12 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.2 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1-3 

1.3 

1.3 

1-3 

1-3 

1-3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.3 

1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 
1.4 

1.4 
1.4 
1.4 
1.4 
1.4 

1.4 

1.5 
1.5 
1.5 

8.719396 

721806 

724204 

726588 

728959 

731317 

733663 

735996 

738317 

740326 

742922 

8.745207 

747479 

749740 

751989 

754227 

756453 

758668 

760872 

763065 

765246 

8.767417 

769578 

771727 

773866 

775995 

778114 

780229 

7823*. 

784408 

786486 

8.788554 

790613 

792662 

794701 

796731 

798752 

800763 

802765 

804858 

806742 

8.808717 

810683 

812641 

814589 

816529 

818461 

820384 

822298 

824205 

826103 

8.827992 

829874 

831748 

833613 

835471 

837321 

839163 

840998 

842825 

844644 

402 

399 

397 

395 

393 

391 

389 

387 

385 

383 

381 

379 

377 

375 

373 

371 

369 

367 

365 

364 
362 
360 
358 
356 

365 
353 
351 
350 
348 
346 
345 
343 
341 
340 
338 
337 
335 
334 
332 
331 
329 
328 
326 
325 
323 
322 
320 
319 
318 
316 
315 
314 
312 
311 
310 
3118 
307 
306 
304 
303 

11.280604 
278194 
275793 
273412 
271041 
268683 
266337 
264004 
261683 
259374 
257078 
11.254793 
252521 
250260 
248011 
245773 
243547 
241332 
239128 1 
236935 
234754 
11.232583 
230422 
228273 
226134! 
224005 
221886 
219778 
217680 
215592 
213514 
11.211446 
209387 
207338 
205299 
203269 
201248 
199237 
197235 
195242 
193258 
11.191283 
189317 
187359 
185411 
183471 
181539 
179616 
177702 
175795 
173897 
11.172008 
170126 
168252 
166387 
164629 
162679 
160837 
159002 
157175 
155356 

05234 
05263 
05292 
05321 
05350 
05379 
05408 
05437 
05466 
|05495 
05524 
05553 
j 05682 

1 05611 
j 05640 
! 05669 

1 05698 
!05727 
05756 
05785 
05814 
05844 
05873 
05902 
05931 
05960 
05989 
06018 
06047 
06076 
06105 
03134 
06163 
06192 
06221 
06250 
06279 
06308 
06337 
06366 
06395 
05424 
06453 
06482 
03511 
06540 
06569 
06598 
06627 
05655 
06685 
06714 
06743 
06773 
06802 
06831 
03860 
06889 
■ 06918 
|06947 
;06976 

99863 

99861 

99860 

99858 

99857 

99855 

99854 

99852 

99851 

99849 

99847 

99846 

99844 

99842 

99841 

99839 

99838 

99836 

99834 

99833 

99831 

99829 

99827 

99826 

99824 

99822 

99821 

99819 

99817 

99815 

99813 

99812 

99810 

99808 

99806 

99804 

99803 

99801 

99799 

99797 

99795 

99793 

99792 

99790 

99788 

99786 

99784 

99782 

99780 

99778 

99776 

99774 

99772 

99770 

99768 

99766 

99764 

99762 

99760 

99758 

99766 

60 

59 

58 

57 

56 

55 

54 

53 

52 

61 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. . 


Cotang. 

| Tang. 

1 N. cos 

N.sine. 

/ 


S3 Degrees. 
















































































F<og. Sines and Tangents. (4°) Natural Sines. 25 


r 

Sine. 

D . 10 

" Cosine. 

D. 10' 

' Tang. 

D . 10' 

Co tang. 

;N. sine 

N. cos 


0 

1 

2 

3 

4 

5 

6 
ri 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

36 
33 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 

52 

53 

54 

55 
66 
57 

68 i 
69 j 
60 

8.843585 
845337 
847183 
848971 
850751 
852525 
854291 
856049 
857801 
859546 
861283 
8.863014 
864738 
866456 
868165 
869868 
871565 
873255 
874938 
876615 
878285 
8.879949 
881607 
883258 
884903 
886542 
888174 
889801 
891421 
893035 
894643 
8.896246 
897842 
899432 
901017 
902596 
904169 
905736 
907297 
908853 
910404 
8.911949 
913488 
915022 
916550 
918073 
919591 
921103 
922610 
924112 
925609 
>.927100 
928587 
930068 
931544 
933015 
934481 
935942 
937398 
938850 
94 J 296 | 

300 
299 
298 
297 
295 
294 
293 
292 
291 
2 0 
288 
287 
286 
285 
284 
283 
282 
281 
279 
279 
277 
276 
275 
274 
273 
272 
271 
270 
269 
268 
267 
266 
265 
264 
263 
262 
261 
260 
259 
258 
257 
257 
256 
255 
254 
253 
252 
251 
250 
249 
249 
248 

247 

248 
245 
244 
243 
243 
242 
241 

9.998941 
998932 
998923 
998914 
998905 
998896 
993887 
998878 
993869 
993860 
998851 
9.998841 
998832 
998823 
998813 
998804 
998795 
998785 
998776 
998766 
998757 
9.998747 
998738 
998728 
998718 
998708 
998699 
998689 
998679 
998669 
998659 
9.998649 
998639 
998629 
998619 
998609 
998599 
998589 
998578 
998568 
998558 
9.998548 
998537 
998527 
998516 
998506 
998495 
998485 
998474 
998464 
998453 
9.998442 
998431 
998421 
998410 
998399 
f 98388 
998377 
998366 
998355 
998344 

1.5 

1.5 

1.5 

1 5 
1-5 

1 -5 

1.6 
1.5 
1-5 
1-5 

1.5 

1 -5 

1 5 

1.6 
1.6 
1.6 
1.6 
1.6 
1.6 
1.6 
1.6 

1.6 
1.6 

1.6 
1.6 

1 -6 
1.6 

1 -6 
1.6 

1 .7 

1.7 
1.7 

1.7 
1.7 

1.7 

1.7 
1.7 
1.7 

1.7 

1 -7 

1.7 
1.7 
1.7 

1.7 

1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 
1.8 

1.8 
1.8 

1.8 
1.8 

8.844644 

846455 

848260 

8501)57 

851846 

853628 

855403 

857171 

858932 

860686 

862433 

8.864173 

865906 

867632 

869351 

871064 

872770 

874469 

876162 

877849 

879529 

8.881202 

882869 

884530 

886186 

887833 

886476 

891112 

892742 

894366 

895984 

8.897696 

899203 

900803 

902398 

903987 

905570 

907147 

908719 

910285 

911846 

8.913401 

914961 

916495 

918034 

919668 

921096 

922619 

924136 

925649 

927156 

3.928658 

930155 

931647 

933134 

934616 

936093 

937565 

939032 

940494 

941962 

302 

801 

299 

298 

29/ 

29 1 
295 
293 

292 
291 
290 
289 
288 
287 
286 
284 
283 
282 
281 
280 
279 
278 
277 
276 
276 
274 
273 
272 
271 
270 
269 
268 

267 
266 
266 
264 
263 
262 
261 
260 
259 

268 
257 
256 
256 
255 
254 
253 
262 
251 
250 
249 
249 
248 
247 
246 
245 
244 
244 
243 

11.155356 
153545 
151740 
149943 
148154 
146372 
144597 
142829 
141068 
139314 
137567 
11.135827 
134094 
132368 
130649 
128936 
127230 
125531 
123838 
122151 
120471 
11.118798 
117131 
115470 
113815 
112167 
110524; 
108888' 
107258 
105634 
104016’ 
11.102404 
100797! 
099197 ! 
097602 
096013 
094430 
092853 
091281 
089715 I 
088154 i 
11.086599 j 
085049 
083505 
081966 
080432 
078904 ; 
077381 
075864 
074361 
072844 1 
11.071342 
069845 
068353 1 
066866 l 
065384 1 
063907 |j 
0624361 
060968 1 
059606 II 
058048 i j 

!0697( 
07005 
07034 
07063 
07092 
07121 
07150 
07179 
07208 
07237 
07266 
07295 
07324 
07353 
07382 
07411 
07440 
07469 
07498 
07527 
07556 
07585 
07614 

1 07643 
j 07672 
07701 
|07730 
07759 
07788 
07817 
07846 
07875 
07904 
07933 
07962 
07991 
08020 
08049 
08078 
08107 
08136 
08165 
08194 
08223 
08252 
08281 
08310 
08339 
08368 
08397 
08426 
08455 
08484 
08513 
08542 
08571 
08600 
08629 
08658 
08687 
08716 

99751 
99754 

99752 
99761 
99748 
99746 
99744 
99742 
99740 
99738 
99736 
99734 
99731 
99729 
99727 
99725 
99723 
99721 
99719 
99716 
99714 
99712 
99710 
99708 
99705 
99703 
99701 
99699 
99696 
99694 
99692 
99689 
99887 
99686 
99683 
99680 
99678 
99676 
99673 
99671 
99668 
99666 
99664 
99661 
99659 
9.9667 
99654 
99652 
99649 
99647 
99644 
99642 
99639 
99637 
99635 
39632 
99630 
9.4627 
99625 
99622 
99619 

60 

59 

58 

57 

56 

65 

54 

53 

62 

61 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. |j 

N. cos. 

'v'.sine. 

/ 





85 Degrees. 







































































26 Log. Sines ami Tangents (5°) Natural Sines. '1 ABLE II. 


t 

Sine. 

D. 10" 

Cosine. 

D. 10" 

Tang. 

D. 10" 

Cotang. 

jN. smc. 

N. cos 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 

51 

52 

53 
64 

55 

56 

57 

58 

59 

60 

8.940296 
941738 
943174 
944606 
946034 
947456 
948874 
950287 
951693 
953100 
954499 
8.955894 
957284 
958370 
960052 
961429 
962801 
964170 
965534 
966893 
968249 
8.969600 
970947 
972289 
973628 
974962 
976293 
977619 
978941 
980259 
981573 
8.982883 
984189 
985491 
986789 
988083 
989374 
990660 
991943 
993222 
994497 
8.995768 
997036 
998299 
999560 
0.000816 
002069 
003318 
004563 
005805 
007044 
9.008278 
009510 
010737 
011962 
013182 
014400 
015613 
016824 
01S031 
019235j 

240 

239 

239 

238 

237 

236 

235 

235 

234 

233 

232 

232 

231 

230 

229 

229 

228 

227 

227 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

214 

214 

213 

212 

212 

211 

211 

210 

209 

209 

208 

208 

207 

206 

206 

205 

205 

204 

203 

203 

202 

202 

201 

201 

9.998344 

998333 

998322 

998311 

998300 

998289 

998277 

998266 

998255 

998243 

998232 

9.998220 

998209 

998197 

998186 

998174 

998163 

998151 

998139 

998128 

998116 

9.998104 

998092 

998080 

998068 

998056 

998044 

998032 

998020 

998008 

997996 

9.997984 

997972 

997959 

997947 

997935 

997922 

997910 

997897 

997885 

997872 

9.997860 

997847 

997835 

997822 

997809 

997797 

997784 

997771 

997758 

997745 

9.997732 

997719 

997706 

997693 

997680 

997667 

997654 

997641 

997628 

997614 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

1.9 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.0 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.1 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 

8.941952 

943404 

944852 

946295 

947734 

949168 

950597 

952021 

953441 

954856 

956267 

8.957674 

959075 

960473 

961866 

963265 

964639 

966019 

967394 

968766 

970133 

8.971496 

972855 

974209 

975560 

976906 

978248 

979686 

980921 

982251 

983577 

8.984899 

986217 

987532 

988842 

990149 

991451 

992750 

994045 

995337 

996624 

8.997908 

999188 

9.000465 

001738 

003007 

004272 

005534 

006792 

008047 

009298 

9.010546 

011790 

013031 

014268 

015502 

016732 

017959 

019183 

020403 

021620 

242 

241 

240 

240 

239 

238 

237 

237 

236 

235 

234 

234 

233 

232 

231 

231 

230 

229 

229 

228 

227 

226 

226 

225 

224 

224 

223 

222 

222 

221 

220 

220 

219 

218 

218 

217 

216 

216 

215 

215 

214 

213 

213 

212 

211 

211 

210 

210 

209 

208 

208 

207 

207 

206 

206 

205 

204 

204 

203 

203 

11.058048 
056596 
055148 
053705 
052266 
050832 
049403 
047979 
046559 
045144 
043733 
11.042326 
040926 
039527 
038134 
036745 
035361 
033981 
032606 
031234j 
029867| 
11.028504 
027145! 
025791 [ 
024440 
0230941 
021762; 
020414 
019079 
017749 
016423 
11.015101 
013783 
012468 
011158 
009851 
008649 
007250 
005955 
004663 | 
003376 
11.002092 
000812 1 
10.999535 ! 
998262 ! 
996993 | 
995728 | 
994466 | 
993208 
991953 
990702 
10.9894541 
988210! 
686969 
985732 
984498 
983268 
9830411 
980817 | 
979597 j 
978380 

08716 

08745 

08774 

08803 

08831 

08860 

08889 

08918 

08947 

08976 

09005 

09034 

09063 

09092 

09121 

09150 

,09179 

09208 

109237 
09266 
09295 
09324 
09353 
09382 
09411 
09440 
09469 
09498 
09527 
09556 
09585 
09614 
09642 
09671 
09700 
09729 
06158 
09787 
09816 
09845 
09874 
09903 
09932 
09961 
09990 
10019 
10048 
10077 
10103 1 
10135 
10164 
10192 
10221 
10250 
10279 
10308 
10337 
10366 
10395 
10424 
10453 

99619 

99617 

99614 

99612 

99609 

99607 

99604 

99602 

99599 

99596 

99594 

99591 

99588 

99586 

99583 

99580 

99578 

99575 

99572 

99570 

99567 

99564 

99562 

99559 

99556 

99553 

99551 

99548 

99545 

99542 

99540 

99537 

99534 

99531 

99528 

99526 

99523 

99520 

99517 

99514 

99511 

99508 

99506 

99503 

99500 

99497 

99494 

99491 

99488 

99485 

99482 

99479 

99476 

99473 

99470 

99467 

99464 

99461 

99458 

99455 

99452 

60 

59 
58 
57 
66 
55 
54 
63 
52 
51 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


tlosiue. • 

Sine. 


Co tang. 


Tang. | 

N. cos. 

N .sine. 

/ 


81 D^trrees. 

































































TABLii II. Log. Sines and Tangents. (6°) Natural Sines. 


0 

1 

2 

3 

4 
B 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 
'16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 
62 

63 

64 
66 
6() 
6'. 
68 
69 
60 

Sine. 

D. 10" 

Cosine. D. 10" 

Tang. D. 10"| Cotang. N. sine 

N. cos. 


9.019235 
020435 
021632 
022825 
024016 
025203 
026386 
027567 
028744 
029918 
031089 
9.032257 
033421 
034582 
035741 
036896 
038048 
039197 
040342 
041485 
042625 
9.043762 
044895 
046026 
047154 
048279 
049400 
050519 
051635 
052749 
053859 
9.054966 
056071 
057172 
058271 
059367 
060460 
061551 
062639 
063724 
064806 
9.065885 
066962 
068036 
059107 
070176 
071242 
072306 
073366 
074424 
075480 
9. (.76533 
077583 
078631 
079676 
980719 
081759 
082797 
083832 
084864 
085894 

200 

199 

199 

198 

198 

197 

197 

196 

196 

195 

195 

194 

194 

193 

192 

192 

191 

191 

190 

190 

189 

189 

180 

188 

187 

187 

186 

186 

185 

185 

184 

184 

184 

183 

183 

182 

182 

181 

181 

180 

180 

179 

179 

179 

178 

178 

177 

177 

176 

176 

175 

175 

175 

174 

174 

173 

173 

172 

172 

172 

9.997614 

997601 

997588 

997574 

997561 

997547 

997634 

997520 

997507 

997493 

997480 

9.997466 

997452 

997439 

997425 

997411 

997397 

997383 

997369 

997355 

997341 

9.997327 

997313 

997299 

997285 

997271 

997257 

997242 

997228 

997214 

997199 

9.997185 

997170 

997156 

997141 

997127 

997112 

997098 

997083 

997068 

997053 

9.997039 

997024 

997009 

996994 

996979 

996964 

996949 

996934 

996919 

996904 

9.996889 

996874 

996858 

996843 

996828 

996812 

996797 

996782 

996766 

996751 

2.2 

2.2 

2.2 

2.2 

2.2 

2.2 

2.3 

2.3 

2-3 

2-3 

2-3 

2-3 

2-3 

2.3 
2-3 
2-3 
2-3 

: 23 

1 2-3 
2-3 
2-3 
2-4 
2-4 

2.4 
2-4 
2-4 
2.4 
2.4 
2-4 
2-4 
2-4 
2-4 
2-4 
2-4 
2-4 
2.4 

2.4 
2-4 
2-5 
2-5 
2-5 
2-6 
2-5 

2.5 
2-5 
2-5 
2-5 
2.5 
2-5 
2.5 

2.5 

2.6 

2.5 

2.6 
2-5 

2.5 

2.6 

2 6 
2.6 
2.6 

9.021620 

022834 

024044 

025251 

026455 

027655 

028852 

030046 

03123/ 

032425 

033609 

9.034791 

035969 

037144 

038316 

039485 

040651 

041813 

042973 

044130 

045284 

9.046434 

047582 

048727 

049869 

051008 

052144 

053277 

054407 

055535 

056659 

9.057781 

058900 

060016 

061130 

062240 

063348 

064453 

065556 

066655 

067762 

9.068846 

069038 

071027 

072113 

073197 

074278 

075356 

076432 

077505 

078576 

9.079644 

080710 

081773 

082833 

083891 

084947 

086000 

087050 

088098 

089144 

202 

202 

201 

201 

200 

199 

199 

198 

198 

19/ 

197 

196 

196 

195 

195 

194 

194 

193 

193 

192 

192 

191 

191 

190 

190 

189 

189 

188 

188 

187 

187 

186 

186 

185 

185 

185 

184 

184 

183 

183 

182 

182 

181 

181 

181 

180 

180 

179 

179 

178 

178 

178 

177 

177 

176 

176 

175 

175 

175 

174 

10.978380 
9/7166 
975956 
974749 
973545 
972345 
9/1148 
969954 
968763 
967575 
966391 
10.965209 
964031 
962856 
961684 
960515 
959349 
958187 
957027 
955870 
954716 
10.953566 
952418 
951273 
950131 
948992 
947856 1 
946723 
945693 
944465 
943341 
10.942219 
941100 
939984 
938870 
937760 
936652 
9S5547 
934444 
933345 
932248 
10.931154 
930062 
928973 
927887 
926803 
925722 
924644 
923568 
922495 
921424 
10-920366 
919290 
918227 
9171671 
9161091 
915053 
914000 
912950 
911902 
910866 

10453 

1 Ji82 

1 511 

10540 
10569 
1059 
10621 
■10655 
10684 
10713 
10742 
j 10771 

i10806 

10829 

10858 

1 10887 
10916 
j10945 
10973 
i11002 
111031 
11060 
j11089 
j 11118 

11147 
11176 
11205 
11234 
11263 
11291 
11320 
11349 
11378 
11407 
11436 
11465 
11494 
11523 
11552 
11580 
11609 
11638 
11667 
11696 
11725 
11754 
11783 
11812 
11840 
11869 
11898 
11927 
11956 
11985 
12014 
12043 
12071 
12100 
12129 
12158 
12187 

09452 

09449 

99446 

99443 

99440 

99437 

99434 

99431 

99428 

99424 

99421 

99418 

“9415 

99412 

99409 

99406 

99402 

99399 

99396 

99393 

99390 

99386 

99383 

99380 

99377 

99374 

99370 

99367 

99364 

99360 

99357 

99354 

99351 

99347 

99344 

99341 

99337 

99334 

99331 

99327 

99324 

99320 

99317 

99314 

99310 

99307 

99303 

99300 

99297 

99293 

99290 

99286 

99283 

99279 

99276 

99272 

99269 

99255 

99262 

99258 

99255 

i 60 
! 59 
58 
67 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
1* 
16 
16 
14 
13 
12 
11 
10 
9 
8 
7 
6 
6 
4 
3 
2 

1 

0 


Cosine. • 

Sine. 


Cotang. 

| Tang. 

N. cos.iN.sine. 

/ 


83 Degrees. 






































































Ix>g. Sines and Tangents. (7°) Natural Sines. TABLE II. 


! 

0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
66 

57 

58 

59 

60 

Sine. 


Cosine. 

D. 10"; Tang. 

D. W'j oOtaug. N. sine. 

S. cos. 


9.085894 
066922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
095056 
096062 
9.097065 
098066 
099065 
100052 
101056 
102048 
103037 
104025 
105010 
105992 
9.106973 
10/951 
108927 
109901 
110873 
111842 
112809 
113774 
114737 
115698 
9.116656 
117613 

118567 
119519 
120469 
121417 
122362 
123306 
124248 
125187 
9.126125 
127060 
127993 
128925 
129854 
130781 
131706 
132630 
133551 
. 134470 
9.135387 
136303 
137216 
138128 
139037 
139944 
140850 
141754 

142655 
143555 

jD. It/ 

171 

171 

170 

170 

170 

169 

169 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

164 

164 

164 

163 

163 

163 
162 
162 
162 
161 
161 
160 
160 
160 
159 
159 
169 
158 
158 
158 
157 
157 
157 
156 
156 
156 
155 
155 
154 

164 
164 
153 
153 
153 
152 
162 
152 
152 
161 
151 
151 
150 
150 

9.993751 
993735 
996720 
996704 
996688 
996673 
996657 
996641 
996625 
996610 
996594 
9.996578 
996562 
996546 
996530 
995514 
996498 
996482 
990465 
996449 
996433 
9.996417 
996400 
996384 
996368 
996351 
996335 
996318 
996302 
996986 
99'.269 
9.996252 
996235 
996219 
* 996202 
996185 
996168 
996151 
996134 
996117 
996100 
9.996083 
996066 
996049 
996032 
996015 
995998 
995980 
995963 
995946 
995928 
9.995911 
995894 
995876 
995859 
995841 
995823 
995806 
995788 
995771 
996753 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.6 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.7 

2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.8 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 
2.9 

9.0S9144 

090187 

091228 

092266 

093302 

094336 

095367 

096395 

097422 

098446 

099468 

9.100487 

101504 

102519 

103532 

104542 

105550 

106556 
107559 
108560 
109559 
9.110556 
111651 
112643 
113533 
114521 
115507 
116491 
117472 
118462 

119429 
9.120404 
121377 
122348 
123317 
124284 
125249 
126211 
127172 
128130 
129087 
9.130641 
130994 
131944 
132893 
133839 
134784 
135726 
136667 
137605 
138642 
9.139476 
140409 
141340 
142269 
143196 
144121 
145044 
145966 
146885 
147803 

174 

173 

173 

173 

172 

172 

171 

171 

171 

170 

170 

169 

169 

169 

168 

168 

168 

167 

167 

166 

166 

166 

165 

165 

165 

164 

164 

164 
163 

163 
162 
162 
162 
161 
161 
161 
160 
160 
160 
159 
169 
159 
158 
158 
158 
157 
157 
157 
156 
156 
156 
155 
155 

165 

164 
154 
154 
153 
153 
153 

10.910356 
909813 
908772 
907734 
906698 
905664 
904633 
903605 
902578 
901554 
900532 
10.899513 
898496 
897481 
896468 
895458 
894450 
893444 
892441 
891440 
890441 
10.889444 
888449 
887457 
886467 
885479 
884493 
883509 
882528 
881548 
880571 
10.879596 
878623 
877652 
876683 
875716 
874751 
873789 
872828 
871870 
870913 
10.869959 
869006 
868056 
867107 
866161 
865216 
864274 
863333 
862395 
861458 
10.860524 
859591j 
858660 
857731 
856804 
855879 
854956 
854G34 
853115 
852197 

12187 
12216 
12245 
12274 
12302 
12331 
12360 
12389 
12418 
12447 
12476 
12504 
12533 
12662 
12591 
12620 
12649 
'12678 
12705 
12736 
12764 
12793 
12822 
12861 
12880 
12908 
12937 
12966 
12995 
13024 
13053 
13081 
13110 
13139 
13168 
13197 
13226 
13254 
13283 
13312 
13341 
13370 
13399 
13427 
13456 
13485i 
13514! 
13543! 
135721 
13600 
13629 
13658 
13687 
13716 
13744 
13773 
13802 
13831 
13860 
13889 
13917 

99255 
99251 
99248 
99244 
99240 
99237 
99233 
99230 
99226 
99222 
99219 
99215 
99211 
99208 
99204 
99200 
99197 
99193 
99189 
99186 
99182 
99178 
99175 
99171 
99167 
99163 
99160 
99156 
99152 
99148 
99144 
99141 
99137 
99133 
99129 
99125 
99122 
99118 
99114 
99110 
99106 
99102 
99098 
99094 
99091 
99087 
99083 
99079 
99075 
99071 
99067 
99063 
99059 
99055 
99051 
99047 
99043 
99039 
99035 
99031 
99027 

60 

59 

58 

57 

56 

55 

64 

53 

•52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

o 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. N’. eos. 

N.sine. 

/ 


82 Degrees. 

■ - ■ - 1 

























































TABLE II. 

I 

jog. Sines 

iml Tangents. (8°) Natural Shu-.-. 


1 

29 


Sine. 

II). 10' 

Cosine. 

1). lu' 

Tang. 

D. hr 

Cotang. 

si ik» 

N . cos 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 
18 

-49 

50 

51 

52 

53 

54 
65 

56 

57 

58 

59 

60 

9.143655 
144453 
145349 
146243 
147136 
148026 
148915 
149802 
150686 
151569 
152451 
9.153330 
154208 
155083 
155957 
156830 
157700 
158569 
159435 
160301 
1*1164 
9.162025 
162885 
163743 
164600 
165454 
166307 
167159 
168008 
168856 
169702 
9.170547 
171389 
172230 
173070 
173903 
174744 
175578 
176411 
177242 
178072 
9.178900 
179726 
180551 
181374 
182196 
183016 
183834 
184651 
185466 
186280 
9.187092 
187903 
188712 
189519 
190325 
191130 
191933 
192734 
193534 
194332 

150 

149 

149 

149 

148 

148 

148 

147 

147 

147 

147 

146 

146 

146 

145 

145 

145 

144 

144 

144 

144 

143 

143 

143 

142 

142 

142 

142 

141 

141 

141 

140 

140 

140 

140 

139 

139 

139 

139 

138 

138 

138 

137 

137 

137 

137 

136 

136 

136 

136 

136 

135 

135 

135 

134 

134 

134 

134 

133 

133 

9.995753 
995735 
995717 
995699 
995681 
995664 
995646 
995628 
995610 
995591 
995573 
9.995555 
995537 
995519- 
995501 
995482 
995464 
995446 
995427 
995409 
995390 
9.995372 
995353 
995334 
995316 
995297 
995278 
995260 
995241 
995222 
995203 
9.995184 
995165 
995146 
995127 
995108 
995089 
995070 
995051 
995032 
995013 
9.994993 
994974 
994955 
994935 | 
994916 j 
994896 
994877 
994857 
994838 
994818 
4.994798 
994779 
994769 
994739 
994719 
994700 
994680 
994660 
994640 
994620| 

3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
i 3.0 
3.0 
3.0 
3.0 
3.0 
3.0 
8.0 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 
3.1 

3.1 

3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 
3.2 

3.2 

3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 
3.3 

9.147803 

148718 

149632 

150544 

151454 

152363 

153269 

154174 

155077 

155978 

156877 

9.157776 

158671 

159565 

160457 

161347 

162236 

163123 

164008 

164892 

165774 

9.166654 

167532 

168409 

169284 

170157 

171029 

171899 

172767 

173634 

174499 

9.175362 

176224 

177084 

177942 

178799 

179655 

180508 

181360 

182211 

183059 

9.183907 

184752 

185597 

186439 

187280 

188120 

188958 

189794 

190629 

191462 

9.192294 

193124 

193953 

194780 

195606 

196430 

197253 

198074 

198894 

199713 

153 

152 

152 

152 

151 

151 

151 

150 

150 

150 

150 

149 

149 

149 

148 

148 

148 

148 

147 

147 

147 

146 

146 

146 

146 

145 

145 

145 

144 

144 

144 

144 

143 

143 

143 

142 

142 

142 

142 

141 

141 

141 

141 

140 

140 

140 

140 

139 

139 

139 

139 

138 

138 

138 

138 

137 

137 

137 

137 

136 

10.852197 
851282 
850368 
849456 
848546 
847637 
846731 
845826 
844923 
844022 
843123 
10 842225 
841329 
840435 
839543 
838653 
837764 
836877 
835992 
835168 
834226 
10-833346 
832468 
831591 
830716 
829843 
828971 
828101 
827233 
826366 
825501 
10.824638 
823776 
822916 
822058 
821201 
820345 
819492 
818640 
817789 
816941 
10-816093 
815248 
814403 
813561 
812720 
811880 
811042 
810206 
809371 
808538! 
10.807706 
806876 
806047 
805220 
804394 
803570| 
802747 | 
801926i 

801106i 

800287 | 

13917 
13946 
13975 
i! 14004 
(! 14033 
14061 
!14090 
j 14119 
i 14148 
14177 
i 14205 
14234 
1 14263 
| 14292 
14320 
,14349 

1 14378 
! 14407 
! 14436 
|14464 
i 14493 
14522 
14661 
14580 
14608 
14637 
14666 
14695 
14723 
14752 
14781 
14810 
14838 
14867 
14896 
14926 
14954 

114982 

1 15011 
15040 
15069 
15097 
15126 
15155 
15184 

1 15212 
i15241 
15270 
15299 
15327 
15356 
15385 
15414 
16442 
15471 
15500 
15529 

15557 

15586 
15615 
15643 

99027 

99023 

99019 

99015 

99011 

99006 

99002 

98998 

98994 

98990 

98986 

98982 

98978 

98973 

98969 

98965 

98961 

98957 

98953 

98948 

98944 

98946 

98936 

98931 

98927 

98923 

98919 

98914 

98910 

98906 

98902 

98897 

98893 

98889 

98884 

98880 

98876 

98871 

98867 

98863 

98858 

98854 

98849 

98845 

98841. 

98836 

98832 

98827 

98823 

98818 

98814 

98809 

98805 

98800 

98796 

9879! 

98787 

98782 

98778 

98773 

98769 

60 

59 

58 

67 

66 

55 

54 

63 

1 62 

61 

50 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

36 

37 , 
36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

16 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. | 

Cotang. | 

Tang. (J N. cos. 

Is.sine 

1 


. 



81 Degrees. 

































































































30 Log. Sines and Tangents. (9°) Natural Sines. TABLE II. 


r 

Sine. 

D. 10" 

Cosine. 

9.994620 

994600 

994580 

994560 

994540 

994519 

994499 

994479 

994459 

994438 

994418 

9.994397 

994377 

994357 

994336 

994316 

994295 

994274 

994254 

994233 

994212 

9.994191 

994171 

994150 

994129 

994108 

994087 

994066 

994045 

994024 

994003 

9.993981 

993960 

993939 

993918 

993896 

993875 

993854 

993832 

993811 

993789 

9.993768 

993746 

993726 

993703 

993681 

993660 

993638 

993616 

993594 

993572 

9.993550 

994528 

993506 

993484 

993462 

993440 

993418 

993396 

993374 

993351 

D. 10 

Tang. 

D. lu' 

Cotang. 

N. sine. 

N. cos. 


0 

1 

2 
o 
O 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 
10 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 
62 
63 

54 

55 

56 

57 

58 

59 

60 

9.194332 

195129 

195925 

196719 

197511 

198302 

199091 

199879 

200666 

201451 

202234 

9.203017 

203797 

204577 

205354 

206131 

206906 

207679 

208452 

209222 

209992 

9.210760 

211526 

212291 

213055 

213818 

214579 

215338 

216097 

216854 

217609 

9.218363 

210116 

219868 

220618 

221367 

222115 

222861 

223606 

224349 

225092 

9.225833 

226573 

227311 

228048 

228784 

229518 

230252 

230984 

231714 

232444 

9.233172 

233899 

234626 

235349 

236073 

230795 

237515 

238235 

238953 

239670 

133 

133 

132 

132 

132 

132 

131 

131 

131 

131 

130 

130 

130 

130 

129 

129 

129 

129 

128 

128 

128 

128 

127 

127 

127 

127 

127 

126 

126 

126 

126 

125 

125 

125 

125 

125 

124 

124 

124 

124 

123 

123 

123 

123 

123 

122 

122 

122 

122 

122 

121 

121 

121 

121 

120 

120 

120 

120 

120 

119 

3.3 

3.3 

3.3 

3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 
3.4 

3.4 

3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 
3.5 

3.5 

3.6 

3.5 

3.6 

3.5 

3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 
3.6 

3.6 

3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 
3.7 

9.199713 

20J529 

201345 

202159 

202971 

203782 

204592 

205400 

206207 

207013 

207817 

9.208619 

209420 

210220 

211018 

211815 

212611 

213405 

214198 

214989 

215780 

9.216568 

217356 

218142 

218926 

219710 

220492 

221272 

222052 

222830 

223600 

9.224382 

225156 

225929 

226700 

227471 

228239 

229007 

229773 

230539 

231302 

9.232066 

232826 

233586 

234346 

235103 

235859 

236614 

237368 

238120 

238872 

9.239622 

240371 

241118 

241866 

242610 

243354 

244097 

244839 

245579 

246319 

136 

136 

136 

135 

135 

135 

135 

134 

134 

134 

134 

133 

133 

133 

133 

133 

132 

132 

132 

132 

131 

131 

131 

131 

130 

130 

130 

130 

130 

129 

129 

129 

129 

129 

128 

128 

128 

128 

127 

127 

127 

127 

127 

126 

126 

126 

126 

126 

125 

125 

125 

125 

125 

124 

124 

124 

124 

124 

123 

123 

10.800287 
799471 
798655 
797841 
797029 
796218 
795408 
794600 
793793 
792987 
792183 
10.791381 
790580 
789/80 
7889821 
7881851 
787389 
786595 j 
785802 
735011 j 
784220 i 
10.783432 
782644 
781858 
781074 
780290 
779508 
778728 | 
777948 
777170 
776394 
10.775618 
774844 
774071 
773300 
772529 
771761 : 
770993 
770227 ! 
769461 j 
768698 | 
10.767935 
767174 
766414 
765655 
764897 
764141 
763386 
762632 
761880 
761128 
10.760378 
759(129 
758882 
758135 
757390 
756646 
755903 
755161 
754421 
753681 

15643 

15672 

15701 

15730 

15758 

16787 

15816 

15845 

15873 

15902 

15931 

15959 

15988 

16017 

16046 

16074 

16103 

16132 

16160 

16189 

16218 

16246 

16275 

16304 

16333 

16361 

16390 

16419 

16447 

16476 

16505 

16533 

16562 

16591 

16620 

16648 

16677 

16706 

116734 
!16763 
!16792 
16820 
16849 
16878 
16906 
16936 
16964 
16992 
17021 
17050 
17078 
17107 
17136 
17164 
17193 
17222 
17250 
17279 
17308 
17336 
17365 

98769 
98764 
98760 
98755 
98751 
98746 
98741 
98737 
98732 
98728 
98723 
98718 
98714 
98709 
98704 
98700 
98695 
98690 
98686 
98681 
98676 
98671 
98667 
98662 
98657 
98652 
98648 
98643 
98638 
98633 
98629 
98624 
98619 
98614 
98609 
98604 
98600 
98595 
98590 
.98585 
98580 
98575 
98570 
98565 
98561 
98556 
98551 
985 46 
98541 
98536 
98531 
98526 
98521 
98516 
98511 
98506 
98501 
98496 
98491 
98486 
98481 

60 
59 
58 
57 
56 
56 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
33 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 ; 
13 | 
12 

|!1 

8 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

/ 


80 Degrees. 





































































TABLE II. Log. Sines and Tangents. (10°) Natural Sines. 


•31 



Sine. 

0 

9.239670 

1 

240386 

2 

241101 

3 

241814 

4 

242526 

5 

243237 

6 

243947 

7 

244656 

8 

245363 

9 

246069 

10 

246775 

11 

9.247478 

12 

248181 

13 

248883 

14 

24958a 

15 

250282 

16 

250980 

17 

251677 

13 

252373 

19 

253067 

20 

253761 

,21 

9.254453 

22 

255144 

23 

255834 

24 

256523 

25 

257211 

26 

257898 

27 

258583 

28 

259268 

29 

259951 

30 

260633 

31 

9.261314 

32 

261994 

33 

262673 

34 

263361 

35 

264027 

36 

264703 

37 

265377 

38 

266051 

39 

266723 

40 

267395 

41 

9.268065 

42 

268734 

43 

269402 

44 

270069 

45 

270735 

46 

271400 

47 

272064 

48 

272726 

49 

273388 

1 60 

274049 

51 

9,274708 

62 

275367 

53 

276024 

54 

276681 

65 

277337 

56 

277991 

57 

278644 

58 

279297 

69 

279948 

60 

280599 


Cosine. | 


D. 10" 


119 

119 

119 

119 

118 

118 

118 

118 

118 

117 

117 

117 

117 

117 

116 

116 

116 

116 

116 

116 

116 

116 

116 

116 

116 

114 

114 

114 

114 

114 

113 

113 

113 

113 

113 

113 

112 

112 

112 

112 

112 

111 

111 

111 

111 

111 

111 

110 

no 

no 

no 

110 

110 

109 

109 

109 

109 

109 

109 

108 


Cosine. 


.993351 

993329 

993307 

993285 

993262 

993240 

993217 

993195 

993172 

993149 

993127 

.993104 

993081 

993059 

993036 

993013 

992990 

992967 

992944 

992921 

992898 

•992875 

992852 

992829 

992806 

992783 

992769 

992736 

992713 

992690 

992666 

.992643 

992619 

992596 

992572 

992549 

992525 

992601 

992478 

992454 

992430 

.992406 

992382 

992359 

992335 

992311 

992287 

992263 

992239 

992214 

992190 

.992166 

992142 

992117 

992093 

992069 

992044 

992020 

991996 

991971 

991947 


D. 10" 


Sine. 


3.7 

3.7 

3.7 

3.7 

3.7 

3.7 

3.8 

3.8 

8.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 
3.8 

3.8 

3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
3.9 
4.0 
4.0 
4.0 
4.0 


4.0 

4.0 

4.0 

4.0 

4.0 

4.0 

4.1 

4.1 

4.1 

4.1 

4.1 

4.1 

4.1 


Tang. 


9. 


.246319 
247057 
247794 
248630 
249264 
249998 
250730 
261461 
252191 
252920 
253648 
.254374 
255100 
265824 
266547 
257269 
257990 
258710 
259429 
260146 
260863 
.261578 
262292 
263005 
263717 
264428 
265138 
265847 
266655 
267261 
267967 
.268671 
269375 
270077 
270779 
271479 
272178 
275876 
273573 
274269 
274964 
.276658 
276351 
277043 
277734 
278424 
279113 
279801 
280488 
281174 
281868 
282542 
283225 
283907 
284588 
285268 
285947 
286624 
287301 
287977 
288652 


Co tang. 


I). 10"| Cotang. 


123 

123 

123 

122 

122 

122 

122 

122 

121 

121 

121 

121 

121 

120 

120 

120 

120 

120 

120 

119 

119 

119 

119 

119 

118 

118 

118 

118 

118 

118 

117 

117 

117 

117 

117 

116 

116 

116 

116 

116 

116 

115 

116 
115 
115 
115 
115 
114 
114 
114 
114 
114 
114 
113 
113 
113 
113 
113 
113 
112 


10 . 


10.753681 
752943 
752206 
751470 
760736 
750002 
749270 
748539 
747809 
747080 
746362 
745626 
744900 
744176 
743453 
742731 
742010 
741290 
740571 
739854 
739137 
10.738422 
737708 
736995 
736283 
735572 
734862 
734153 
733445 
732739 
732033 
10.731329 
730625 
729923 
729221 
728521 
727822 
727124 
726427 
725731 
725036 
’0.724342 
723649 
722957 
722266 
721576 
720887 
720199 
719612 
718826 
718142 
10.717458 
716775 
716093 
715412 
714732 
714053 
713376 
712699 
712023 
711348 


N.sine.lN. cos. 


Tang. 


17365 

17393 

17422 

17461 

17479 

17508 

17537 

17566 

17594 

17623 

17651 

17680 

17708 

17737 

17766 

17794 

17823 

17852 

17880 

17909 

17937 

17966 

17996 

18023 

18052 

18081 

18109 

18138 

18166 

18195 

18224 

18252 

18281 

18309 

18338 

18367 

18396 

18424 

18452 

18481 

18609 

18538 

18567 

18595 

18624 

18652 

18681 

18710 

18738 

18767 

18795 

18824 

18852 

18881 

18910 

18938 

18967 

18995 

19024 

19052 

19081 


98481 

98476 

98471 

98466 

98461 

98455 

98450 

98445 

98440 

98435 

98430 

98425 

98420 

98414 

98409 

98404 

98399 

98394 

98389 

98383 

98378 

98373 

98368 

98362 

98367 

98352 

98347 

98341 

98336 

98331 

98326 

98320 

98315 

98310 

98304 

98299 

98294 

98288 

98283 

98277 

98272 

98267 

98261 

98256 

98250 

98245 

98240 

98234 

98229 

98223 

98218 

98212 

98207 

98201 

98196 

98190 

98186 

98179 

98174 

98168 

98163 


l N. cos. N.nrie, 


60 

59 
68 
67 
66 
65 
54 
63 
62 
51 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


79 Degrees. 
































































f=— 

I 32 


Log. Sines and Tangents. (11°) Natural Sines. 

TA 

BLE II. 

i 

Sine. I 

). 10' 

Cosine. ID. lo''! 

Tang. 

J. iu 

v or.aug. j 

S. s,ne. 

-N. COS.| 


1 ° 

1 2 

I 6 

1 8 

1 9 

1 10 
11 

1 12 

1 13 

1 14 

1 15 

1 16 
17 

1 18 

1 19 
20 

21 

1 22 

1 23 

1 24 
25 

1 26 
27 

1 28 

1 29 
1 3U 
1 31 

1 32 
33 

I 34 
I 35 
1 36 
37 
I 38 
I 39 
I 40 

1 41 

1 42 

I 43 

I 44 
45 
I 46 
[ 47 
48 
I 49 
50 

1 51 

52 

53 
I 54 
I 55 
1 56 

1 57 

58 

59 

60 

).280599 
281248 
281897 
282544 
283190 
283836 
284480 
285124 
285766 
286408 
287048 
9.287687 
288326 
288964 
289600 
290236 
290870 
291504 
292137 
292768 
293399 
9.294029 
294658 
295286 
295913 
296539 
297164 
297788 
298412 
299034 
299655 
9.300276 
300895 
• 301514 
302132 
302748 
303364 
303979 
304593 
305207 
305819 
9.306430 
307041 
307650 
308259 
308867 
309474 
310080 
310685 
311289 
311893 
9.312495 
313097 
313698 
314297 
314897 
315495 
316092 
316689 
317284 
317879 

198 * 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
106 
106 
106 
106 
106 
105 
105 
105 
105 
105 
105 
104 
104 
104 
104 
104 
104 
104 
103 
103 
103 
103 
103 
103 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
99 
99 
99 

).991947 
991922 
991897 
991873 
997848 
991823 
991799 
991774 
991749 
991724 
991699 
9.991674 
991649 
991624 
991599 
991574 
991549 
991524 
991498 
991473 
991448 
9.991422 
991397 
991372 
991346 
991321 
991295 
991270 
991244 
991218 
991193 
9.991167 
991141 
991115 
991090 
991064 
991038 
991012 
990986 
990960 
990934 
S 990908 
990882 
990855 
990829 
990803 
990777 
990-50 
990724 
990697 
990671 
9.990644 
990ol8 
990591 
990565 
990538 
990511 
990485 
990458 
990431 
990404 

4.1 

4.1 

4.1 

4.1 

4.1 

4.1 

4.1 

4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 
4.2 

4.2 

4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 
4.3 

4.3 

4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 
4.4 

4.4 
4.6 
4.6 
4.6 

4.5 

1.288652 
2-9326 
289999 
290671 
291342 
292013 
292682 
293350 
294017 
294684 
295349 
9.296013 
296677 
297339 
298001 
298662 
299322 
299980 
300638 
301295 
301951 
9.302607 
303261 
303914 
304567 
305218 
305869 
306519 
307168 
307815 
308463 
9.309109 
309754 
310398 
311042 
311685 
312327 
312967 
313608 
314247 
314885 
9.315523 
316159 
316795 
317430 
318064 
318697 
319329 
319961 
320592 
321222 
9.321851 
322479 
323106 
323733 
324358 
324983 
325607 
326231 
326853 
327475 

112 

112 

112 

112 

112 

111 

111 

111 

111 

111 

111 

111 

110 

110 

no 

no 

no 

no 

109 

109 

109 

109 

109 

109 

109 

108 

108 

108 

108 

108 

108 

107 

107 

107 

107 

107 

107 

107 

106 

106 

106 

106 

106 

106 

106 

105 

105 

105 

105 

105 

105 

105 

104 

104 

104 

104 

104 

104 

104 

104 

10.711348! 
710674 
710011 i 
709329 |! 
708658 
707987 
707318 
706650 
705983 
705316 
704651 
10.7039871 
703323 
702661 
701999 
701338 
700678 
700020 
699362 
698705 
698049 
10-697393 
696739 
696086 
695433 
694782 
694131 
693481 
692832 
692185 
691537 
10-690891 
690246 
689602 
688958 
688315 
687673 
687033 
686392 
685753 
685116 
10-684477 
683841 
683205 
682570 
681936 
681303 
680671 
680039 
679408 
678778 
10-678149 
677521 
676894 
676267 
675642 
675017 
674393 
673769 
673147 
672525 

19081 
19109! 
19138 
19167 
19195 
19224 
19252 
19281 
19309 
19338 
19366 
19395 
19423 
19452 
19481 
19509 
19538 
19566 
19595 
19623 
19652 
19680 
19709 
19737 
19766 
19794 
19823 
19851 
19880 
19908 
19937 
19965 
19994 
20022 
20051 
20079 
20108 
20136 
20165 
20193 
20222 
20250 
20279 
20307 
20336 
20364 
20393 
120421 
20450 
20478 
2050- 
' 20535 
; 20563 
I 20592 
| 20520 
i 20649 
1 20677 

1 20706 
i 20734 
I 20763 
1 20791 

78163 

78157 

38152 

16146 

48140 

48135 

98129 

98124 

98118 

98112 

98107 

98101 

98096 

98090 

98084 

98079 

98073 

98067 

98061 

98056 

98050 

98044 

98039 

98033 

98027 

98021 

98016 

98010 

98004 

97998 

97992 

97987 

97981 

97975 

97969 

97963 

97958 

97952 

97946 

97940 

97934 

97928 

97922 

97916 

97910 

97905 

97899 

97893 

97887 

97881 

97875 

97869 

97863 

97857 

97851 

97845 

9-839 

97833 

97827 

97821 

97815 

60 

59 

58 

67 

66 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotaug. 


Tang. 

1 N. c os 

. N.sine 

I t 

\ 





78 Degrees 




• 

! 










































































1 


TABLE II. Log. Sines and Tangents. 

D. 10'' Cosine 


(12°) Natural Sines. 


33 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 
4!) 

50 

51 

52 

53 

54 

55 

56 
67 

58 

59 

60 


9 


9 


9 


9. 


Sine. 

.317879 
318473 
319)6, 
319658 
320249 
320840 
321430 
322019 
322607 
323194 
323780 
,324366 
324950 
325534 
326117 
326700 
327281 
337862 
328442 
329021 
329599 
.330176 
330753 
33132J) 
331963 
332478 
333051 
333624 
334195 
334766 
335337 
.335905 
336475 
337043 
337610 
338176 
338742 
339306 
339871 
340434 
340996 
.341558 
342119 
342679 
343239 
343797 
344355 
344912 
345469 
346024 
346579 
347134 
Q47687 
348240 
348792 
349343 
349893 
3 >0443 
350992 
351540 
352088 


99.0 

98.8 

98.7 


Cosine. 


98 
98 
98 
98 
98. 
97. 
97. 
97. 
97. 
97. 
97. 
97. 
96. 
96. 

96.6 

96.5 

98.4 

96.2 

96.1 
96.0 

95.8 

95.7 

95.6 

95.4 

95.3 

95.2 
95.0 

94.9 

94.8 

94.6 

94.5 

94.4 

94.3 

94.1 
94.0 

93.9 

93.7 

93.6 

93.5 

93.4 

93.2 

93.1 
93.0 

92.9 

92.7 

92.6 

92.5 

92.4 

92.2 
92.1 
92.0 

91.9 

91.7 

91.6 

91.5 
91.4 
91.3 


9 


9 


9 


9 


.990104 
990378 
990351 
999324 
999297 
999270 
999243 
990215 
990188 
990181 
990134 
.990107 
990079 
990052 
990025 
989997 
989970 
989942 
989915 
989887 
989860 
.989832 
989804 
989777 
989749 
989721 
989693 
989665 
989637 
989609 
989582 
.989553 
989525 
989497 
989469 
989441 
989413 
989384 
989356 
989328 
989300 
.989271 
989243 
989214 
989186 
989157 
989128 
989100 
989071 
989042 
989014 
.988985 
988956 
988927 
988898 
988869 
988840 
988811 
988782 
988753 
988724 


Sine. 


D. 10" 


4.5 
4.5 
4.5 
4 5 
4.5 
4.5 

4.5 

4.6 
4.6 
4.5 

4.5 

4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 
4.6 

4.6 

4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 
4.7 

4.7 

4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 
4.8 

4.8 

4.9 
4.9 
4.9 


Tan- 


9.327474 
328095 
328715 
329334 
329953 
330570 
331187 
331803 
332418 
333033 
333646 
9.334259 
334871 
335482 
336093 
336702 
337311 
337919 
338527 
339133 
339739 
9.340344 
340948 
341552 
342155 
342757 
343358 
343958 
344558 
345157 
345755 
9.346353 
346949 
347545 
348141 
348735 
349329 
349922 
350514 
351106 
351697 
9.352287 
352876 
353465 
354053 
354640 
355227 
355813 
356398 
356982 
357566 
9.358149 
358731 
359313 
359893 
360474 
361053 
361632 
362210 
362787 
363364 


Co tang. 


L>. to 

103 

103 

103 

103 

103 

103 

103 

102 

102 

102 

102 

102 

102 

102 

102 

101 

101 

101 

101 

101 

101 

.’01 

lOl 

10U 

100 

100 

100 

100 

100 

100 

100 

99.4 
99.3 

99.2 

99.1 
99.0 

98.8 
98.7 

98.6 

98.5 

98.3 

98.2 

98.1 
98.0 

97.9 

97.7 

97.6 

97.6 

97.4 

97.3 

97.1 
97.0 

96.9 

98.8 

96.7 
96.6 

96.5 

96.3 

96.2 
96.1 


Coiaug. 

N. sme. iN. cos. 


10.672526 

20791 97815 

60 

671905) 

20820 97809 

59 

671285| 

20848 97803 

58 

670866 

20877 97797 

67 

670047 

20905 97791 

56 

669430 

20933 97784 

55 

668813 

20962 97778 

54 

668197 

209.90 97772 

53 

667582 

21019 97766 

52 

666967 

21047 97760 

51 

666354 

21076 97754 

50 

10.665741 

21104 9774-> 

49 

665129 

21132 97742 

48 

664518 

21161 97735 

47 

663907 

21189 97729 

46 

663298 

21218 97723 

45 

662689 

21246 97717 

44 

662081 

21275 97711 

43 

661473 

21303 97705 

42 

660867 

21331 97698 

41 

660261 

21360 97692 

40 

10.659656 

21388 97686 

39 

659052 

21417 97680 

38 

658448 

21445 97673 

37 

657845 

21474 97667 

36 

657243 | 

21502 97661 

35 

656642 

21530 97655 

34 

656042 

21559 97648 

33 

655442 

2158797642 

32 

654843 

21616 97636 

31 

654245! 

21644 97630 

30 

10.653647 

21672 976^3 

29 

653051 

21701 97617 

28 

652455 

21729 97611 

27 

651859 

21758 97604 

26 

651265 

21786)97598 

25 

650671 

21814:97592 

2* 

650078 

2184397585 

23 

649486 

21871 97579 

22 

648894 

21899 97573 

21 

648303 

21928,97566 

20 

10.647713 

21956 97560 

19 

647124; 

21985 97553 

18 

646535 

22013197547 

17 

645947 

i 22041 97541 

16 

645360 

! 2207 0 97534 

15 

644773! 

122098197528 

14 

6441871 

2212697521 

13 

643602 J 

22155 97515 

12 

643018 1 

22183 97508 

11 

642434 

122212 97602 

10 


10 


641851 

641269 

640687 

640107 

639526 

638947 

638368 

637790 

637213 

636636 


2224097496 
i 22268197489 
122297 97483 
22325 97476 
22353 97470 


Tang. 


22382 
122410 
;22438 
J 22467 
22495 


97463 

97457 

97450 

97444 

97437 


N. cos. N.sine. 


9 

8 

7 

6 

5 

4 

3 

o 

1 

0 


Lsir* : 


Degrees. 
















































































34 


Log. Sines and Tangents. (.13°) Natural Sines. 


TABLE II. 


1 

Sine. 

D. lu" 

Cosine. 

0 

9.352088 

91.1 
91.0 
90.9 

90.8 

90.7 

90.5 

90.4 
90.3 

90.2 

90.1 

89.9 

89.8 

89.7 

89.6 

89.5 

89.3 

89.2 

89.1 
89.0 

88.9 

88.8 

88.7 

88.6 

88.4 

88.3 

88.2 
88.1 
88.0 

87.9 

87.7 

87.6 

87.5 

87.4 

87.3 

87.2 

87.1 
87.0 

86.9 

86.7 

86.6 

86.5 

86.4 

86.3 

86.2 
86.1 
86.0 

85.9 

85.8 
85.7 

85.6 

QR A 

9.988724 

1 

352635 

9386y5 

2 

353181 

988666 

3 

353726 

988636 

4 

354271 

988607 

5 

354815 

988578 

6 

355358 

988548 

7 

355901 

988519 

8 

356443 

988489 

9 

356984 

988460 

10 

357524 

988430 

11 

9.358064 

9.988401 

12 

358603 

988371 

13 

359141 

988342 

14 

359678 

988312 

15 

360215 

988282 

16 

360752 

988252 

17 

361287 

988223 

18 

361622 

988193 

19 

362356 

988163 

20 

362889 

988133 

21 

9.363422 

9.988103 

22 

363954 

988073 

23 

364486 

988043 

24 

365016 

988013 

25 

365546 

987983 

26 

366076 

987953 

27 

366604 

987922 

28 

367131 

987892 

29 

367659 

987862 

30 

368185 

987832 

31 

9.368711 

9.987801 

32 

369236 

987771 

33 

369761 

987740 

34 

370286 

987710 

35 

370808 

987679 

36 

371330 

987649 

37 

371852 

987618 

38 

372373 

987688 

39 

372894 

987567 

40 

373414 

987526 

41 

9.373933 

9.987496 

42 

374452 

987465 

43 

374970 

987434 

44 

376487 

987403 

45 

376003 

987372 

46 

376519 

987341 

47 

377036 

987310 

48 

377649 

987279 

49 

378063 

987248 

50 

378577 

987217 

51 

9 ,379069 

85.3 

86.2 

85.1 

85.0 

84.9 

84.8 

84.7 

84.6 

84.5 

9.987186 

52 

63 

379601 

380113 

987155 

987124 

54 

380624 

987092 

55 

381134 

987061 

56 

381643 

987030 

57 

382152 

986998 

58 

382661 

986967 

59 

383168 

986936 

60 

383675 

986904 


Cosine. 


Sine. 


D. 10'' 


4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

4.9 

5.0 

6.0 

5.0 

5.0 

5.0 

6.0 

5.0 

6.0 

6.0 

5.0 

6.0 

6.0 

6.0 

6.0 

5.0 

5.0 

5.1 

6.1 
5.1 
5.1 
5.1 
5.1 
5.1 
5.1 

5.1 

6.1 
5.1 
5.1 


Tang. 


L>. lo' 


5.2 

5.2 

5.2 

5.2 

6.2 
5.2 
5.2 
5.2 
5.2 
5.2 
5.2 

5.2 

6.2 
6.2 
6.2 


9.303364 
363940 
364615 
365090 
365664 
366237 
366810 
367382 
367963 
368524 
369094 
9.369663 
370232 
370/99 
371367 
371933 
372499 
373064 
373629 
374193 
374756 
9.375319 
375881 
376442 
377003 
377563 
378122 
378681 
379239 
379797 
380364 
380910 
381466 
382020 
382575 
383129 
383682 
384234 
384786 
385337 
385888 
9.386438 
386987 
387536 
388084 
388631 
389178 
389724 
390270 
390815 
391360 
9.391903 
392447 
392989 
393531 
394073 
394614 
395154 
395694 
396233 
396771 


96.0 

95.9 

95.8 

95.7 

95.5 

95.4 

95.3 

95.2 

95.1 
95.0 

94.9 

94.8 

94.6 

94.5 

94.4 

94.3 

94.2 

94.1 
94.0 

93.9 

93.8 

93.7 

93.5 

93.4 

93.3 

93.2 

93.1 
93.0 

92.9 

92.8 

92.7 

92.6 
y2 6 
92 4 

92.3 

92.2 

93.1 

92.0 

91.9 

91.8 

91.7 

91.5 

91.4 

91.3 

91.2 

91.1 
91.0 

90.9 

90.8 

90.7 

90.6 

90.6 

90.4 

90.3 

90.2 
90.1 
90.0 

89.9 

89.8 

89.7 


Cotang. 


Cotang. 

N.sine 

N. cos. 


10.636636 

22495 

97437 

60 

636060 

22523 

97430 

69 

635485 ! 

22552 

97424 

58 

634910 

22580 

97417 

67 

634336 

22608 

97411 

66 

633763 

22637 

97404 

66 

633190 

22665 

97398 

54 

632618 

22693 

97391 

53 

632047 

22722 

97384 

62 

631476 

22760 

9,378 

51 

630906 

22778 

97371 

60 

10.630337 

22807 

97365 

49 

629768 

22835 

97358 

48 

629201 

22863:97351 

47 

628633 

22892 

97345 

46 

628067 

22920 

97338 

45 

627501 

22948 

97331 

44 

626936 

22977 

97325 

43 

626371 

23005 

97318 

42 

625807 

23033 

97311 

41 

625244 

23062 

97304 

40 

10.624681 

23090 

97298 

39 

624119 

23118 

97291 

38 

623558 

23146 

97284 

37 

622997 1 

23175 

97278 

36 

622437 

23203 

97271 

36 

621878 

23231 

97264 

34 

621319 

23260 

97257 

83 

620761 

23288 

97261 

32 

620203 

23316 

97244 

31 

619646 j 

•^3345 

97237 

30 

10.619090| 

23373 

91230 

29 

618534 | 

23401 

97223 

28 

617980! 

23429 

97217 

27 

617425 i 

23468 

9,210 

26 

616871 1 

23486 

97203 

25 

616318 

23514 

9719b 

24 

615766 

23542 

97169 

23 

615214 

235 71 

97182 

22 

614663 

23599 

97176 

21 

614112 

23627 

97169 

20 

10.613562 

23656 

97162 

19 

613013 

23684 

97155 

18 

612464 

23712 

97148 

17 

611916 

23740 

97141 

16 

611369 

23769 

97134 

15 

610822 

23797 

97127 

14 

610276 

23825 

97120 

13 

609730 

23853 

97113 

12 

609185 

23882 

9710., 

11 

609640 

23910 

97100 

10 

10.608097 

23938 

97093 

9 

607553 

23966 

970~>6 

8 

607011 

23995 

(97079 

7 

606469 

24023 

970.2 

6 

605927 

24051 

97066 

5 

605386 

240,9 

9 7 05 , 

4 

604846 

24108 

97051 

3 

604306 

24l3t, 

97044 

2 

603767 

24164 

97037 

1 

603229 

24192 

97030 

0 

Tang 

! N. cos.jN.sine 

"i 


76 Degrees. 








































































1 


TABLE II. Log. Sines and Tangents. (14°) Natural Sines. 35 

0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
501 
611< 
62 

63 

64 

65 

66 

67 

68 
69 
60 

bme. 

^ .. 

D. 10' 

Cosine. 

D. lo' 

' Tang. 

D. 10' 

Cotang. 

IN. sine 

. N. cos 

• 

9.383675 
384182 
384687 
385192 
385697 
386201 
386704 
387207 
387709 
388210 
388711 
9.389211 
389711 
390210 
390708 
391208 
391703 
392199 
392695 
393191 
393685 
9.394179 
394673 
395166 
395658 
396150 
396641 
397132 
397621 
398111 
398600 
9 399088 
399575 
400062 
400549 
401035 
401520 
402005 
402489 
402972 
403455 

9 403938 
404420 
404901 
405382 
405862 
406341 
406820 
407299 
407777 
408254 

9 408731 
409207 
409682 
410157 
410632 
411106 
411579 
412052 
412524 
412996 

84.4 

84.3 

84.2 

84.1 
84.0 
83.9 
83.8 

83.7 

83.6 

83.5 

83.4 

83.3 

83.2 

83.1 
83.0 

82.8 

82.7 

82.6 

82.5 

82.4 

82.3 

82.2 
82.1 
82.0 

81.9 

81.8 
81.7 

81.7 

81.6 
81.6 

81.4 

81.3 
81.2 
81.1 
81.0 

80.9 

80.8 

80.7 
80.6 
80.6 

80.4 

80.3 
80.2 
80.1 
80.0 

79.9 

79.8 

79.7 

79.6 

79.5 

79.4 
79.4 
79.3 
79.2 
79.1 
79.0 

78.9 

78.8 

78.7 

78.6 

9.986904 
9.86873 
986841 
98(5809 
936778 
986746 
986714 
986683 
986651 
986619 
986587 
9.986555 
986523 
986491 
98645 9 
986427 
936395 
986363 
986331 
986299 
986266 
9 986234 
9862(12 
986169 
986137 
986104 
986072 
986039 
986007 
985974 
985942 
9 985909 
985876 
985843 
985811 
985778 
985745 
985712 
985679 
985646 
985613 
9.985580 
985547 
985514 
985480 
985447 
985414 
985380 
985347 
985314 
985280 

9 985247 
985213 
985180 
985146 
986113 
985079 
985045 
985011 
984978 
984944 

5.2 

5.3 
5.3 

5.3 

6.3 

5.3 

6.3 

5.3 

5.3 

6.3 

5.3 
5.3 
5.3 

5.3 

6.3 

6.3 

5.3 

6.4 

6.4 

5.4 

6.4 

5.4 

6.4 

6.4 

5.4 

5.4 

6.4 

5.4 

6.4 

5.4 

5.4 

5.5 

5.5 

6.5 

5.5 

5.5 

5.6 
5.6 
5.6 
5.6 

5.6 

6.6 
5.6 

5.6 

6.6 
5.6 
5.6 
5.6 

5.6 

6.6 

5.6 

6.6 
6.6 

5.6 

6.6 
6.6 
6.6 
5.6 
5.6 
5.6 

9.396771 
397309 
397846 
398383 
398919 
399455 
39.6990 
400524 
401058 
401591 
402124 
9.402656 
403187 
403718 
404249 
404778 
406308 
405836 
406364 
406892 
407419 
9 407945 
408471 
408997 
409521 
410045 
410569 
411092 
411615 
412137 
412668 
9.413179 
413699 
414219 
414738 
415257 
415775 
416293 
416810 
417326 
417842 

9 418368 
418873 
419387 
419901i 
420415 
420927 
421440 
421962 
422463 
422974 
) 423484 
423993 
424603 
425011 
426519 
426027 
426534 
427041 
427647 
428052 

89.6 

89.6 
89.5 

89.4 

89.5 

89.2 

89.1 
89.0 

88.9 
88.8 

88.7 

88.6 

88.5 

88.4 

88.3 

88.2 
88.1 
88.0 

87.9 

87.8 

87.7 

87.6 

87.5 

87.4 

87.4 

87.3 

87.2 

87.1 
87.0 

86.9 

86.8 

86.7 

86.6 

86.5 

86.4 

86.4 

86.3 

86.2 
86.1 
86.0 

85.9 

86.8 

85.7 

85.6 

86.5 

85.6 

85.4 

86.3 
86.2 
85.1 
85.0 

84.9 

84.8 
84.8 
8 *. 7 

84.6 
84.6 

84.4 
84.3 
84.3- 

10.603229 
602691 
602154 
601617 
601081 
600545 
600010 
599476 
698942 
698409 
597876 
10 697344 
596813 
596282 
695751 
595222 
594692 
594164 
593636 
693108 
592581 
10.592055 
691529 
691003 i 
590479! 
689955| 
589431 
688908 
688385 
687863 
687342 
10-586821 
586301 
585781 
685262 
684743 
684225 
683707[ 
683190 
682674 
682158 
0.681642 
681127 
580613 
680099 
679585 
679073 
678660 
678048 ' 
677537 1 
677026 j 
10.576616; 
676007 
676497 
674989i 
674481 | 
673973 
578466; 
672959 
672453 | 
571948 ! 

2419S 
2422t 
24246 
24277 
2430c 
24332 
24362 
2439C 
j24418 
[24446 
1 24474 
;24503 
24531 
j24559 
124587 
24615 
24644 
24672 
24700 
24728 
24756 
24784 
24813 
24841 
24869 
24897 
24925 
24954 
24982 
25010 
26038 
25066 
25094 
25122 
25161 
26179 
25207 
25235 
25263 
25291 
25320 
25348 
25376 
25404 
26432 
25460 
25488 
25516 
25545 
25573 
26601 
25629 
25657 
25685 
25713 
25741 
25766 
25798 
25826 
25854 
25S82 

97030 
97023 
197015 
97008 
97001 
96994 
96987 
96980 
96973 
96966 
c96969 
96952 
96945 
96937 
96930 
96923 
96916 
96909 
96902 
96894 
96887 
96880 
96873 
96866 
96858 
96851 
96844 
96837 
96829 
96822 
96816 
96807 
96800 
96793 
96786 
96778 
96771 
96764 
96766 
96749 
96742 
96734 
96727 
96719 
96712 
96705 
96697 
96690 
96682 
96675 
96667 
96660 
96653 
96645 
96638 
96630 
96623 
96615 
96608 
96600 
96593 

60 

59 
58 
57 
56 
55 
64 
63 
62 
61 

60 
49 
48 
47 
46 
46 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

20 

19 

18 

17 

16 

15 

14 

13 i 
12 1 

11 

l V 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Co tang. 


Tang. i i N. cos. 

N.sine. 

t 


75 Degrees. 





















































































36 


Log. Sines and Tangents. (15°; Katural Sines. TABLE II. 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 

52 

53 
64 
66 
66 
57 
68 
69 
60 


9. 


Sine. 


D. 10" 


9. 


412996 
413467 
413938 
414408 
4148/8 
415347 
415S15 
416283 
416751 
417217 
417684 
418150 
418615 
419079 
419544 
420007 
420470 
420933 
421395 
421857 
422318 
.422778 
423238 
4236*7 
424156 
424615 
425073 
425530 
425987 
426443 
426899 
.427354 
427809 
428263 
428717 
429170 
429623 
430076 
430527 
430978 
431429 
.431879 
432329 
432778 
433226 
433676 
434122 
434569 
435016 
435462 
435908 
.436353 
436798 
437242 
437686 
438129 
438572 
439014 
439456 
439897 
440338 


78.5 
78.4 
78.3 

78.3 

78.2 

78.1 
78.0 
77.9 

77.8 

77.7 

77.6 

77.6 

77.4 

77.3 

77.3 

77.2 

77.1 
77.0 

76.9 

76.8 

76.7 

76.7 

76.6 

76.5 

76.4 

76.3 

76.2 

76.1 
76.0 
76.0 

75.9 

75.8 

75.7 

75.6 

76.5 

76.4 

75.3 

75.2 

75.2 

75.1 
760 

74.9 
74.9 

74.8 

74.7 
74 6 

74.5 

74.4 

74.4 

74.3 

74.2 
74.1 
74.0 
74.0 

73.9 

73.8 
73.7 

73.6 
73 6 

73.5 


9. 


9. 


Cosine. 


D. 10" Tang. 


Cosine. 


984944 
984910 
984876 
984842 
984808 
984774 
984740 
984706 
984672 
984637 
984603 
984669 
981535 
984500 
984466 
984432 
984397 
984363 
984328 
984294 
984259 
,984224 
984190 
984155 
984120 
984085 
984060 
984016 
983981 
983946 
983911 
.983875 
983840 
983805 
983710 
983735 
983.00 
983664 
983629 
983594 
983658 
.983623 
983487 
983452 
983416 
983381 
983345 
983309 
983273 
983238 
983202 
.983166 
983130 
983094 
983058 
983022 
982986 
982950 
982914 
982878 
982842 


5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

5.7 

C.7 


9. 


Sine. 


5.8 

5.8 

5.8 

6.8 
6.8 
5.8 
5.8 
5.8 
5.8 

5.8 

6.8 

5.8 

6.8 
6.8 

5.8 

6.8 

5.9 

6.9 
6.9 
6.9 

6.9 

5.9 
5.9 
5.9 

5.9 

6.9 
6.9 

6.9 

5.9 
5.9 
5.9 
5.9 
6.0 
6.0 
6.0 
6.0 
6.0 
6.0 
6#0 
6.0 
6.0 
6.0 
6.0 
6.0 


9. 


9. 


428062 
42855 7 
429062 
429565 
430070 
430573 
431075 
431577 
432079 
432580 
433080 
433580 
434080 
434579 
435078 
435576 
436073 
436670 
437067 
437563 
438059 
438664 
439048 
439543 
440036 
440529 
441022 
441614 
442006 
442497 
442988 
.443479 
443968 
444458 
444947 
445436 
445923 
446411 
446898 
447384 
447870 
.448356 
448841 
449326 
449810 
460294 
450777 
451260 
451743 
452225 
462706 
.453187 
463668 
454148 
454628 
456107 
456686 
456064 
466542 
457019 
457496 


to. io" 

84.2 

84.1 
84.0 
83.9 
83.8 

83.8 

83.7 

83.6 

83.5 
83.4 

83.3 

83.2 

83.2 

83.1 
83.0 

82.9 

82.8 
82.8 

82.7 

82.6 
82.6 

82.4 

82.3 

82.3 

82.2 
82.1 
82.0 

81.9 
81 .9 

81.8 

81.7 
81.6 
81.6 

81.5 

81.4 

81.3 
81.2 
81.2 
81.1 
81.0 

80.9 

80.9 

80.8 

80.7 

80.6 
80.6 

80.5 

80.4 
80.3 
80.2 
80.2 
80.1 
80.0 

79.9 
79.9 

79.8 
79.7 

79.6 
79.6 

79.5 


LOtul) r 


N. sine 


10 . 


10 


10 


10 


10 


Cotang. 


10 


571948 
571443 
670938 
670434 
669930 
669427 
558925 
668423 
667921 
667420 
566920 
566420 
665920 
665421 
664922 
564424 
663927 
663430 
662933 
562437 
561941 
,561446 
660952 
660457 
559964 
659471 
658978 
558486 
657994 
657503 
557012 
.556521 
556032 
655542 ; 
655053| 
654565 I 
554077| 
553589 | 
553102 
652616 
652130 
.561644 
651159 
650674 
650190 
649706 
649223 
648740 
648267 
5477/5 
547294 
.546813 
646332 
645852 
645372 
544893 
644414 
643936 
643458 
642981 
642504 


N. cos. 

96593 
96585 
96678 
96570 
96562 
96555 
96547 
96540 
96532 

_ 96624 

26163 96517 


25882 

25910 

25935 

2596s> 

25994 

2602" 

26050 

26079 

2610/ 

26135 


26191 

26219 

26247 

26275 

26303 

26331 


96509 

96502 

96494 

96486 

96479 

96471 


26359 96463 
26387 96456 


26415 
26443 
26471 
26500 
26528 
26556 
26584 
26612 
26640 
26668 
26696 
26724 
26752 
26780 
26808 
26836 
26864 
26892 
26920 
26948 
269/6 
27004 
27032 
2/ 060 
27088 
27116 
27144 


96448 

96440 

96433 

96425 

96417 

96410 

96402 

96394 

96386 

96379 

96371 

96363 

96355 

96347 

96340 

96332 

96324 

96316 

96308 

96301 

96293 

96285 

96277 

96269 

96261 

96263 

96246 


27172196238 
27200 96230 
27228 96222 
27256 96214 
27284196206 
27312 96198 
27340 96190 


27368 

27396 

27424 

27452 

27480 

27508 

27536 

27564 


I Tang. 


96182 

96174 

96166 

96158 

96150 

96142 

96134 

96126 


N. cos. N.sine. 


60 

59 

68 

57 

56 

55 

54 

53 

52 

31 

60 
49 
48 
47 
46 
46 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 

32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 

8 

7 

6 

6 

4 

o 

O 

2 

1 

0 


74 Degrees. 













































































r 


TABLE II. Log. Sines and Tangents. (16°) Natural Sines. 37 


/ 

Sine. 

D. 10" 

Cosine. 

D. 10" 

Tang. 

D. 10" 

Cotang. 

N. sine. 

N. cos. 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.410338 
440778 
441218 
441658 
442096 
442535 
442973 
443410 
443847 
444284 
444720 
9.445155 
445590 
446025 
446459 
446893 
447326 
447759 
448191 
448623 
449054 
9.449485 
449915 
450345 
450775 
451204 
451632 
452060 
452488 
452915 
453342 
9.463768 
454194 
454619 
455044 
456469 
456893 
456316 
456739 
457162 
457584 
9 458006 
458427 
458848 
459268 
459688 
460108 
46052 7 
460946 
431364 
461782 
3.462199 
462616 
463032 
463448 
463864 
464279 
464694 
465108 
465522 
465935 

73.4 

73.3 

73.2 
73-1 

73.1 
73.0 
72.9 

72.8 
72.7 

72.7 

72.6 

72.5 

72.4 

72.3 

72.3 

72.2 

72.1 
72.0 
72.0 

71.9 

71.8 

71.7 

71.6 
71.6 

71.6 

71.4 

71.3 

71.3 

71.2 

71.1 
71.0 
71.0 

70.9 

70.8 

70.7 

70.7 
70.6 

70.5 

70.4 

70.4 

70.3 

70.2 

70.1 
70.1 
70.0 

69.9 

69.8 

69.8 
69.7 

69.6 

69.5 

69.6 

69.4 

69.3 
69.3 

69.2 
69.1 
69.0 
69.0 

68.9 

9.982842 
982805 
982769 
982733 
982696 
982660 
982624 
982587 
982551 
982514 
982477 
9.982441 
982404 
982367 
982331 
982294 
982257 
982220 
982183 
982146 
982109 
9 982072 
982035 
981998 
981961 
981924 
981886 
981849 
981812 
981774 
981737 
9.981699 
981662 
981625 
981687 
981649 
981512 
981474 
981436 
981399 
981361 
9.981323 
981285 
981247 
981209 
981171 
981133 
981095 
981057 
981019 
980981 
9.980942 
980904 
980866 
980827 
980789 
980750 
980712 
980673 
980636 
980596 

6.0 
6.0 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.1 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.2 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 
6.3 

6.3 

6.4 
6.4 
6.4 
6.4 
6.4 

6.4 

6.4 

6.4 
6.4 
6.4 
6.4 
6.4 
6.4 
6.4 

9.457496 
457973 
458449 
458925 
459400 
459875 
460349 
460823 
461297 
461770 
462242 
9 462714 
463186 
463658 
464129 
464699 
465069 
466539 
466008 
466476 
466946 
9 46/413 
467880 
468347 
468814 
469280 
469746 
470211 
470676 
471141 
471605 
9 472068 
'472632 
472996 
473457 
473919 
474381 
474842 
475303 
475763 
476223 
9 476683 
'477142 
477601 
478069 
478517 
478975 
479432 
479889 
480345 
480801 
9 481257 
481712 
482167 
482621 
483076 
483529 
483982 
484435 
484887 
485339 

79.4 
79.3 

79.3 
79 2 

79.1 
79.0 
79.0 
78.9 

78.8 

78.9 

78.7 
78-6 
78-5 

78.5 

78.4 
78 3 

78.3 

78.2 

78.1 
78.0 
78-0 
77-9 
77-8 

77.8 
77-7 
77-6 
77-6 
77-6 
77-4 
77-3 
77-3 

77.2 
77.1 

77.1 
77.0 

76.9 
76.9 

76.8 
76.7 

76.7 

76.6 

76.5 

76.6 

76.4 

76.3 

76.3 

76.2 
76.1 
76.1 
76.0 

76.9 

75.9 

76.8 

76.7 

75.7 
76.6 

76.5 
76.5 

75.4 
76 .'3 

10.642504 

542027 

541551 

541075 

640600 

640125 

639651 

639177 

638703 

638230 

637758 

10.537286 

636814 

636342 

635871 

636401 

634931 

634461 

633992 

633624 

533055 

10.532587 

532120 

531653 

631186 

630720 

630254 

629789 

629324 

628859 

628396 

10.527932 

627468 

627005 

626643 

626081 

526619 

625168 

624697 

524237 

623777 

10.623317 

522858 

522399 

521941 

621483 

621025 

520568 

520111 

619665 

619199 

10.618743 

618288 

617833 

617379 

516926 

616471 

516018 

516565 

615113 

614661 

27564 

27592 

27620 

27648 

27676 

27704 

27731 

27759 

27787 

27815 

27843 

27871 

27899 

27927 

27955 

27983 

28011 

28039 

28067 

28095 

28123 

28150 

28178 

28206 

28234 

28262 

28290 

28318 

28346 

28374 

28402 

28429 

28467 

28486 

28613 

28641 

28669 

28597 

28626 

28652 

28680 

28708 

28736 

28764 

28792 

28820 

128847 

28876 

28903 

28931 

28959 

28987 

29015 

29042 

29070 

29098 

29126 

29164 

29182 

29209 

29247 

96126 
96118 
96110 
96102 
96094 
96086 
96078 
96070 
96062 
96054 
96046 
96037 
96029 
96021 
96013 
9()005 
95997 
95989 
95981 
969 J 2 
95964 
95956 
95948 
95940 
95931 
95923 
95915 
95907 
95898 
95890 
95882 
96874 
95865 
95867 
95849 
95841 
95832 
95824 
96816 
35807 
95799 
95791 
95782 
95774 
95766 
96767 
95749 
95740 
95732 
95724 
95715 
95707 
95698 
95690 
95681 
95673 
95664 
95656 
95647 
95639 
95630 

60 

59 

68 

67 

66 

66 

64 

63 

52 

61 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

*2 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

/ 

73 Degrees. 

























































r 


38 Log. Sines and Tangents. (17°) Natural Sines. TABLE II. 


/ 

Sine. 

D. 10' 

Cosine. 

D. 10' 

Tang. 

D. 1U' 

Cotang. | N. sine 

.[N. cos 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 
63 

54 

55 

56 

57 

58 

59 

60 

9.465935 

466348 

466761 

467173 

467585 

467996 

468407 

468817 

469227 

469637 

470046 

9.470455 

470863 

471271 

471679 

472086 

472492 

472898 

473304 

478710 

474115 

9.474519 

474923 

475327 

475730 

476133 

476536 

476938 

477340 

477741 

478142 

9.478542 

478942 

479342 

479741 

480140 

480539 

480937 

481334 

481731 

482128 

9.482525 

482921 

483316 

483712 

484107 

484501 

484895 

485289 

485682 

486075 

9.486467 

486860 

487251 

487643 

488034 

488424 

488814 

489204 

489593 

489982 

68.8 

68.8 

68.7 
68.6 
68.5 

68.5 

68.4 
68.3 

68.3 
68.2 
68.1 
68.0 
68.0 
67.9 

67.8 

67.8 

67.7 

67.6 

67.6 

67.5 

67.4 

67.4 

67.3 
67.2 

67.2 

67.1 
67.0 

66.9 

66.9 

66.8 

66.7 

66.7 

66.6 
66.6 

66.5 

66.4 

66.3 

66.3 

66.2 
66.1 
66.1 
66.0 

65.9 
65.9 

65.8 
65.7 

65.7 

65.6 
65.6 

65.5 

65.4 

66.3 

65.3 
66.2 
65.1 
65.1 
65.0 
65.0 

64.9 

64.8 

9.980596 

1 980558 
i 980519 
980480 
980442 
980403 
980364 
980325 
980286 
980247 
980208 
9.980169 
980130 
980091 
980052 
980012 
979973 
979934 
979895 
979855 
979816 
9.979776 
979737 
979697 
979658 
979618 
979579 
979539 
979499 
979459 
979420 
9.979380 
979340 
979300 
979260 
979220 
979180 
979140 
979100 
979059 
979019 
9.978979 
978939 
978898 
978858 
978817 
978777 
978736 
978696 
978655 
978615 
9.978574 
978533 
978493 
978452 
978411 
978370 
978329 
978288 
978247 
978206 

6.4 

6.4 

6.5 
6.5 
6.5 
6.5 
6.5 
6.5 
6.5 

6.5 

6.6 

6.5 

6.6 
6.6 
6.6 
6.5 

6.5 

6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.6 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 
6.7 

6.7 

6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 
6.8 

9.485339 

485791 

486242 

486693 

487143 

487593 

488043 

488492 

488941 

489390 

489838 

9.490286 

490733 

491180 

491627 

492073 

492519 

492965 

493410 

493854 

494299 

9.494743 

495186 

495630 

496073 

496515 

496957 

497399 

497841 

468282 

498722 

9.499163 

499603 

500042 

500481 

600920 

601359 

601797 

602235 

602672 

603109 

9.503646 

603982 

604418 

604854 

605289 

505724 

606159 

606593 

607027 

607460 

9.607893 

608326 

51*8759 

509191 

509622 

610054 

610485 

610916 

511346 

611776 

75.3 

75.2 

76.1 

75.1 
75.0 
74.9 
74.9 

74.8 
74.7 

74.7 
74.6 
74.6 

74.6 

74.4 

74.4 

74.3 

74.3 

74.2 

74.1 
74.0 
74.0 
74.0 

73.9 

73.8 

73.7 

73.7 
73.6 

73.6 

73.5 

73.4 
73.4 

73.3 

73.3 

73.2 
73.1 

73.1 
73.0 
73.0 

72.9 

72.8 

72.8 

72.7 

72.7 

72.6 
72.6 
72.6 

72.4 
72.4 

72.3 

72.2 
72.2 
72.1 
72.1 
72.0 

71.9 
71.9 

71.8 
71.8 

71.7 
71.6 

10.514661 
614209 
513768 
613307 
512857 
612407 
611957 
611508 
511059 
610610 
610162 
10.609714 
609267 
608820 
508373 
607927 
607481 
607035 
506590 
606146 
605701 
10.506257 
504814 
604370 
603927 
503485 
603043 
602601 
602159 
601718 
601278 
10.500837 
600397 
499958 
499519 
499080 
498641 
498203 
497765 
497328 
496891 
10.496454 
496018 
495582 
495146 
494711 
494276 
493841 
493407 
492973 
492540 
10.492107 
491674 
491241 
490809 
490378| 
489946 
489515 
489084 
488654 
488224 

I'29237 
29265 
29293 
!29321 
(29348 
29376 
j 29404 
!29432 
j 29460 
[29487 
(29515 
29543 
(29571 
29599 
29626 
29654 
29682 
29710 
29737 
29765 
29793 
29821 
29849 

129876 
29904 
29932 
29960 
2998/ 
30015 
30043 
30071 
30098 
30126 
30154 
30182 
30209 
3023 7 
30265 
30292 
30320 
30348 
30376 
30403 
30431 
30459 
30486 
30514 
30542 
30570 
3059/ 
30625 
30653 
30680 
30708 
30736 
30763 
30791 
30819 
30846 
30874 
30902 

95630 

95622 

95613 

95605 

95596 

95588 

95579 

95571 

95562 

[95554 

95546 

95536 

95528 

95519 

95511 

95502 

95493 

95485 

95476 

95467 

95459 

95450 

95441 

95433 

95424 

95415 

95407 

95398 

95389 

95380 

95372 

95363 

95354 

95345 

95337 

95328 

95319 

95310 

95301 

95293 

95284 

95275 

95266 

95257 

95248 

95240 

95231 

95222 

95213 

95204 

95195 

95186 

95177 

95168 

95159 

^5150 

95142 

95133 

95124 

95116 

95106 

60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

i 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sinc. 

f 

72 Degrees. 































































TABLE IT. Log. Sin/-s and Tangents. (18°) Natural Sines. 


39 


Sine. 


0 
1 
2 

3 

4 
6 
6 

7 

8 
9 

10 

11 9 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 
62 

63 

64 
55 
66 

67 

68 
69 
60 


.489982 
490371 
490759 
491147 
491535 
491922 
492308 
492695 
493081 
493466 
493851 
.494236 
494621 
495005 
495388 
495772 
496154 
496537 
496919 
497301 
497682 
.498064 
498444 
498825 
499204 
499584 
499963 
500342 
600721 
501099 
501476 
.501854 
502231 
602607 
502984 
503360 
503735 
604110 
604485 
504860 
605234 
.505608 
605981 
606354 
506727 
60/099 
607471 
507843 
508214 
608585 
508956 
.609326 
509696 
610065 
510434 
510803 
511172 
511540 
511907 
612276 
612642 


D. 10' 


Cosine. 


64.8 

64.8 

64.7 
64.6 

64.6 

64.5 
64.4 

64.4 

64.3 
64.2 

64.2 
64.1 

64.1 
64.0 

63.9 

63.9 

63.8 

63.7 

63.7 

63.6 

63.6 

63.5 

63.4 

63.4 

63.3 

63.2 

63.2 
63.1 

63.1 
63.0 

62.9 

62.9 

62.8 
62.8 

62.7 

62.6 
62.6 

62.5 

62.5 

62.4 

62.3 

62.3 

62.2 
62.2 
62.1 
62.0 
62.0 

61.9 
61.9 

61.8 
61.8 
61.7 

61.6 
61.6 

61.5 
61.5 

61.4 
61.3 
61.3 
61.2 


Cosine. 


9.978206 
978165 
978124 
978083 
978042 
978001 
977959 
977918 
977877 
977835 
977794 
9.977752 
977711 
977669 
977628 
977586 
977544 
977503 
977461 
977419 
977377 
9.977335 
977293 
977251 
977209 
977167 
977125 
977083 
977041 
976999 
976957 
976914 
976872 
976830 
976787 
976745 
976702 
976660 
976617 
976574 
976532 
9.976489 
976446 
976404 
976361 
976318 
976275 
976232 
976189 
976146 
976103 
976060 
976017 
975974 
975930 
975887 
975844 
975800 
975/67 
975714 
975670 


Sine. 


D. 10' 


6.-8 
6.8 
6.8 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
6.9 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.0 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 
7.1 

7.1 

7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 
7.2 


Tang. 


9.511776 
612206 
6i2635 
513064 
613493 
613921 
614349 
514777 
515204 
615631 
516057 
9.516484 
616910 
517335 
617761 
618185 
618610 
619034 
619458 
519882 
520305 
9.520728 
521151 
621673 
521995 
522417 
622838 
623259 
523680 
524100 
524520 
9.524939 
625359 
525778 
526197 
626615 
527033 
627451 
627868 
528286 
528702 
529119 
629536 
529950 
530366 
630781 
631196 
531611 
532025 
532439 
632863 
9.533266 
633679 
634092 
534604 
634916 
535328 
535*39 
536150 
636561 
636972 


D. 10" 


Cotang. 


71.6 

71.6 

71.5 
71.4 

71.4 
71.3 

71.3 
71.2 

71.2 

71.1 
71.0 
71.0 
70.9 
70.9 
70.8 

70.8 

70.7 

70.6 

70.6 

70.5 

70.5 

70.4 

70.3 
70.3 

70.3 

70.2 

70.2 
70.1 

70.1 
70.0 

69.9 
69.9 

69.8 

69.8 

69.7 

69.7 

69.6 
69.6 

69.6 

69.5 

69.4 

69.3 
69.3 

69.3 

69.2 
69.1 
69.1 
69.0 
69.0 

68.9 
68.9 

68.8 
68.8 

68.7 
68.7 

68.6 
68.6 

68.5 
68.5 

68.4 


Cotang. N. sine. N. cos. 


10 - 


■ >1 


10.488224 
487794 
487365 
486936 
486507 
486079 
485651 
486223 
484796 
484369 
483943 
10.483516 
483090 
482665 
482239 
481815 
481390 
480966 
480542 
480118 
479695 
479272 
478849 
478427 
478005 
477583 
477162 
476741 
476320 
475900 
475480 
10-475061 
474641 
474222 
473803 
473385 
472967 
472549 
472132 
471715 
471298 
10-470881 
470466 
470050 
469634 
46921? 
468804 
468389 
467976 
467661 
467147 
10-466734 
466321 
465908 
465496 
465084 
464672 
464261 
463850 
463439 
463028 


30902 

30929 

30957 

309^5 

31012 

31040 

31068 

31095 

31123 

31151 

31178 

31206 

31233 

31261 

31289 

31316 

31344 


31399 


95106 

95097 

95088 

95079 

95070 

95061 

95052 

95043 

95033 

95024 

95015 

95006 

94997 

94988 

94979 

94970 

94961 


31372 94952 


94943 


3142/194933 
31454194924 
31482 94915 
31510 94906 


31537 

31565 

31593 


94897 
94888 
94878 
31620 94869 


31648' 

31675 

31703 

31730 


94860 

94851 

94842 

94832 


31758 94823 


31786 

31813 

31841 


94814 

94805 

94795 


3186894786 
3189694777 
3192394768 
319511947 58 
31979 94749 
32006194740 
32034 94730 


32061 

32089 


94721 

94712 


Tang. 


32116 94702 
32144'94693 
3217194684 


32199 

32227 


94674 

94665 


32250 94656 
32282 94646 


32309 

32337 

32364 

32392 

32419 

32447 

32474 

32502 

32529 

32557 


94637 

94627 

94618 

94609 

94599 

94590 

94580 

94571 

94561 

94552 


N. cos. N.sine. 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


71 Degrees. 








































































r 


40 Log. Sines and Tangents. (19°) Natural Sines. TABLE II. 


/ 

Sin 

L>. lo' 

Cosine. 

D. io 

Tang 

D. io 

Co tang. 

hN . sine.) X. cos 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

9.512642 
513>09 
613375 
613741 
514107 
614472 
514837 
515202 
615566 
615930 
616294 
9.616657 
517020 
617382 
517745 
518107 
618468 
518829 
519190 
619551 
519911 
9.520271 
520631 
620990 
521349 
521707 
522066 
622424 
622781 
523138 
623495 
9.523852 
524708 
524664 
524920 
625275 
525630 
525984 
526339 
526693 
627046 
9.527400 
627753 
628105 
528458 
628810 
529161 
629613 
629864 
630215 
630565 
9.530915 
531265 
631614 
631963 
632312 
632661 
633009 
633357 
633704 
634052 

61.2 

61.1 

61.1 

61.0 

60.9 

60.9 
60.8 
60.8 
60.7 

60.7 
60.6 

60.5 
60-5 
60.4 

60.4 
60-3 
60-3 
60-2 
60-1 
60.1 
60.0 
60.0 

59.9 
59.9 

69.8 

59.8 

59.7 

59.6 

69.6 

59.5 

59.6 
59-4 

59.4 

69.3 

59.3 
59-2 
59-1 
69-1 
69-0 
59-0 
58-9 

68.9 

68.8 

58.8 

68.7 
58 7 
58-6 
58-6 
58.6 

68.5 
58-4 
58-4 
68-3 
58-2 
58-2 
58.1 
58.1 
58.0 
58.0 

57.9 

9.975670 

97567? 

975583 

975539 

976496 

975452 

975408 

975365 

975321 

976277 

975233 

9.975189 

975145 

975101 

975057 

975013 

974969 

974925 

974880 

974836 

974792 

9.974748 

974703 

974659 

974614 

974570 

974525 

974481 

974436 

974391 

974347 

9.974302 

974257 

974212 

974167 

974122 

974677 

974032 

973987 

973942 

973897 

9.973852 

973807 

973761 

973716 

9736/1 

973625 

973580 

973536 

973489 

973444 

9.973398 

973352 

973307 

973261 

973215 

973169 

973124 

973078 

973032 

972986 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.3 

7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.4 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 
7.6 

7.6 

7.7 

9.536972 
637382 
637792 
538202 
638611 
639020 
639429 
639837 
640245 
640653 
641061 
9.641468 
541875 
642281 
642688 
643094 
643499 
643905 
544310 
644715 
645119 
9.645524 
645928 
646331 
646735 
647138 
647640 
647943 
548345 
648747 
649149 
9.549560 
649951 
650352 
650752 
651162 
661662 
651962 
552351 
662760 
553149 
9.563548 
653946 
564344 
554741 
656139 
655536 
666933 
556329 
666726 
657121 
9.557617 
657913 
658308 
658702 
559097 
659491 
•659885 
660279 
560673 
561066 

68.4 

68.3 

68.3 
68.2 
68.2 
68.1 
68.1 
68.0 
68.0 
67.9 
67.9 
67.8 

67.8 
67.7 

67.7 
67.6 
67.6 
67.6 

67.6 

67.4 

67.4 
67.3 

67.3 
67.2 

67.2 
67.1 

67.1 
67.0 
67.0 

66.9 
66.9 

66.8 
66.8 

66.7 

66.7 
66.6 
66.6 

66.5 

66.6 
66.6 

66.4 

66.4 

66.3 
66.3 

66.2 
66.2 
66.1 
66.1 
66.0 
66.0 
66.9 

66.9 

65.9 

65.8 
65.8 
65.7 
65.7 
65.6 
65.6 

65.5 

10.463028 
462618 
462208 
461798 
461389 
460980 
460571 
460163 
459755 
459347 
458939 
10.458532 
458125 
457719 
457312 
456906 
456501 
456095 
455690 
455285 
454881 
10.454476 
464072 
453669 
453265 
452862 
452460 
452057 
451656 
451253 
450851 
10.450460 
450049 
449648 
449248 | 
448848) 
448448| 
448048 
447649i 
447250 j 
446851i 
10.446452| 
446054 
445656 1 
445259 
444861j 
444464 
444067 11 
443671 
443275 
442879| 
10.442483! 
442087 
441692 
441298 
440903 
440509 
440115| 
439721 |! 
4393271 
438934! 

• 3255" 
3258- 
32615 
32631 
32667 
32694 
32722 
32749 
32777 
32804 
32832 
32859 
32887 
32914 
32942 
32969 
l32997 
33024 
33051 
330/9 
33106 
33134 
33161 
33189 
33216 
33244 
33271 
33298 
33326 
33353 
33381 
33408 
33436 
33463 
33490 
33518 
33545 
33573 
33600 
33627 
33655 
33682 
83710 
33737 
33764 
33792 
33819 
33846 
33874 
33901 
33929 
33956 
33983 
34011 
34038 
34065 
34093 
34120 
34147 
34175 
34202 

94552 
194542 
94533 
94523 
94514 
94504 
94495 
94485 
94476 
94466 
94457 
94447 
94438 
94428 
94418 
94409 
94399 
94390 
94380 
94370 
94361 
94351 
94342 
94332 
94322 
94313 
94303 
94293 
94284 
94274 
94264 
94254 
94245 
94236 
94225 
94215 
94206 
94196 
94186 
94176 
94167 
94157 
94147 
94137 
94127 
94118 
94108 
94098 
94088 
94078 
)4068 
94058 
14049 
94039 
94029 
14019 
94009 
93999 
93989 
13979 
.13969 

60 

69 

58 

67 

56 

55 

54 

53 

62 

51 

60 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

A 

b 

6 

4 

3 

2 

1 

0 
“ 7~ 


Cosine. 


Sine. 

Cotang. 


Tang. I 

N. o/s. 

N’.sine. 


70 Degrees. 



































































Sines and Tangents. (20°) Natural Sines. 41 


t ' 

Sine. 

D. 10' 

' Cosine. 

D. 10' 

Tang. 

D. 10" 

Colaug. 

N. sine 

N. cos 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 
61 
52 
63 

54 

55 

56 

57 

58 
69 
60 

9.534052 
534399 
534 745 
635092 
635438 
535783 
536129 
536474 
636818 
537163 
537507 
9.537851 
638194 
638538 
638880 
639223 
539565 
539907 
640249 
640590 
540931 
9.541272 
641613 
641953 
542293 
642632 
642971 
543310 
643649 
543987 
644326 
9.544663 
645000 
645338 
645674 
646011 
646347 
646683 
547019 
647354 
647689 
9.548024 
648359 
548693 
649027 
549360 
649693 
650026 
650359 
550692 
651024 
9.651356 
651687 
662018 
652349 
652680 
653010 
653341 
653670 
654000 
654329 

57.8 
57.7 

57.7 
57-7 
57.6 
57.6 

57.6 
57.4 

57.4 
57.3 

57.3 
57.2 

57.2 

67.1 

57.1 
57.0 
57.0 

56.9 
56.9 

66.8 

56.8 

56.7 

56.7 
56.6 

56.6 

56.5 

56.5 

66.4 

56.4 

56.3 

56.3 

56.2 

56.2 
56.1 

56.1 
56.0 
56.0 

55.9 
55.9 

55.8 

55.8 

55.7 
55.7 

55.6 
55.6 

55.6 

65.6 

55.4 
55.4 

55.3 
55.3 

55.2 

55.2 

65.2 
55.1 
55.1 
55.0 
55.0 

54.9 

64.9 

9.972986 

972940 

972894 

972848 

972802 

972765 

972709 

972663 

972617 

972570 

972524 

9.972478 

972431 

972385 

972338 

972291 

972245 

972198 

972151 

972106 

972058 

9.972011 

971964 

971917 

971870 

971823 

971776 

971729 

971682 

971635 

971588 

9.971540 

971493 

971446 

971398 

971351 

971303 

971256 

971208 

971161 

971113 

9.971066 

971018 

970970 

970922 

970874 

970827 

970779 

970731 

970683 

970635 

9.970586 

970538 

970490 

970442 

970394 

970345 

970297 

970249 

970200 

970152 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.7 

7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 
7.8 

7.8 

7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
7.9 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.0 
8.1 
8.1 
8.1 
8.1 

9.561066 

661459 

661851 

662244 

562636 

563028 

663419 

663811 

664202 

564592 

664983 

9.565373 

665763 

666163 

666542 

566932 

667320 

567709 

668098 

668486 

568873 

9.669261 

669648 

670036 

570422 

670809 

571196 

671581 

571967 

572362 

572738 

9.673123 

673607 

673892 

674276 

674660 

576044 

576427 

676810 

676193 

676576 

9.576958 

677341 

677723 

578104 

678486 

578867 

579248 

679629 

680009 

680389 

9.680769 

681149 

681528 

681907 

682286 

682665 

683043 

583422 

683800 

584177 

65.6 
65.4 
65.4 
65 3 
65.3 

65.3 
66.2 

65.2 
66.1 

65.1 
65.0 
65.0 
64.9 
64.9 
64.9 
64 8 

64.8 

64.7 

64.7 
64.6 
64.6 
64.6 
64.6 

64.6 

64.4 

64.4 

64.3 

64.3 

64.2 
64.2 

64.2 
64.1 

64.1 
64.0 
64.0 

63.9 
63.9 
63.9 

63.8 

63.8 

63.7 
63.7 
63.6 
63.6 
63.6 
63.6 

63.5 

63.4 
63.4 
63.4 

63.3 
63.3 

63.2 
63.2 
63.2 
63.1 
63.1 
63.0 
63.0 

62.9 

10.438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
436798 
435408 
435017 
10.434627 
434237 
433847 
433458 
433068 
432680 
432291 
431902 
431514 
431127 
10.430739 
430352 
429966 
429678 
429191 
428805 
428419 
428033 
427648 
427262 
10.426877 
426493 
426108! 
426724 
425340 
424956 
424573 
424190 
423807 
423424 
10.423041 
422659 
422277 
421896 | 
4215141 
421133) 
420762| 
420371! 
419991 
419611 
10.419231 
418851 
418472 
418093 
417714 
417335 
416967 
416578 
416200 
416823 

34202 
134229 
134257 
134284 
'34311 
!34339 
34366 
|34393 
34421 
34448 
34476 
34503 
34530 
34557 
34584 
34612 
34639 
34666 
34694 
34721 
34748 
84776 
34803 
34830 
34857 
34884 
34912 
34939 
34966 
34993 
35021 
36048 
35076 
35102 
35130 
36157 
35184 
35211 
36239 
36266 
36293 
35320 
35347 
35375 
35402 
35429 
35456 
35484 
35511 
35538 
35565 
35592 
35619 
35647 
35674 
35701 
35728 
36755 
35782 
35810 
35837 

93969 

93959 

93949 

93939 

93929 

93919 

93909 

93899 

93889 

93879 

98869 

93859 

93849 

93839 

93829 

93819 

93809 

93799 

93789 

93779 

93769 

93769 

93748 

93738 

93728 

93718 

93708 

93698 

93688 

93677 

93667 

93657 

93647 

93637 

93626 

93616 

93606 

93596 

93685 

93675 

93565 

93555 

98544 

93534 

93524 

93514 

93503 

93493 

93483 

93472 

93462 

93452 

93441 

93431 

93420 

93410 

93400 

93389 

93379 

93368 

93368 

60 

59 

58 

67 

66 

55 

54 

63 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

16 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. | 


Tang. 

N. cos. 

N.sine. 

t 


C9 Degrees. 




























































42 Log. Sines and Tangents. (21°) Natural Sines. ' TABLE II . 


/ 

| Sine. 

D. 10* 

Cosine. 

D. 10" 

Tang. 

D. 10*| Cotang. 

|N -sine 

N. cos. 


0 

9.554329 

54.8 

54.8 
54.7 

54.7 
54.6 

54.6 
54.5 

54.5 
54.4 

54.4 
54.3 
54.3 

54.3 
54.2 

54.2 
54.1 

54.1 
54.0 
54.0 

53.9 
53.9 

53.8 

53.8 

53.7 

53.7 

53.6 
53.6 

53.6 

53.5 

53.5 

53.4 

53.4 

53.3 

53.3 

53.2 

53.2 
53.1 
53.1 

53.1 
53.0 
53.0 

52.9 
52.9 

52.8 
52.8 
52.8 

52.7 
52.7 

52.6 
52.6 
52.6 

52.5 

52.4 
52.4 

52.3 
52.3 
52.3 

52.2 
52.2 
52.1 

9.970152 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.1 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

8.2 

Q Cl 

9.584177 

62.9 

62.9 
62.8 
62.8 
62.7 
62.7 

62.7 
62.6 
62.6 
62.5 
62.5 

62.5 
62.4 

62.4 
62.3 
62.3 

62.3 
62.2 
62.2 
62.2 
62.1 
62.1 
62.0 
62.0 

61.9 

61.9 

61.8 
61.8 
61.8 
61.7 
61.7 

61.7 

61.6 
61.6 
61.6 

61.5 
61.5 

61.5 

61.4 
61.4 
61.3 
61.3 
61.3 
61.2 
61.2 
61.1 
61.1 
61.1 
61.0 
61.0 
61.0 

60.9 
60.9 
60.9 

60.8 
60.8 
60.7 
60.7 
60.7 

60.6 

10.415823 

35837 

93358 

60 

1 

554658 

970103 

584555 

415445 

35864 

93348 

59 

2 

654987 

970055 

584932 

415068 

35891 

93337 

58 

3 

555315 

970006 

585309 

414691 

■35918 

93327 

57 

4 

555643 

969957 

585686 

414 j 14 

35945 

93316 

56 

5 

655971 

969909 

586062 

413938 

35973 

93306 

55 

6 

553299 

969860 

586439 

413561 

136000 

93295 

54 

7 

556626 

969811 

586815 

413185 

36027 

93285 

53 

8 

9 

556953 

657280 

969762 

969714 

587190 

587566 

412810 

412434 

' 360.54 
36081 

93274 

93264 

52 

51 

10 

557606 

969665 

587941 

412059 

36108 

93253 

50 

It 

9.557932 

9.969616 

9.588316 

10.411684 

36135 

93243 

49 

12 

558258 

969567 

588691 

411309 

36162 

93232 

48 

13 

558583 

969518 

589066 

410934 

36190 

93222 

47 

14 

558909 

969469 

689440 

410560 

36217 

93211 

46 

15 

659234 

969420 

689814 

410186 

36244 

93201 

45 

16 

559558 

969370 

590188 

409812 

36271 

93190 

44 

17 

559883 

969321 

590562 

409438 

36298 

93180 

43 

18 

560207 

969272 

590935 

409065 

36325 

93169 

42 

19 

560531 

969223 

o .V 
8.2 
8.2 
8.2 
8.2 
8.2 
8.2 
8.3 
8.3 
8.3 
8.3 
8.3 
8.3 

ti Q 

591308 

408692 

j 36352 

93159 

41 

20 

560855 

969173 

691681 

408319 

36379 

93148 

40 

21 

9.561178 

9.969124 

9.592054 

10.407946 

136406 

93137 

39 

22 

561501 

969075 

592426 

407574 

!36434 

93127 

38 

23 

531824 

969025 

592798 

407202 

136461 

93116 

37 

24 

562146 

968976 

593170 

406829 

36488 

93106 

36 

25 

562468 

968926 

593542 

406458 

36515 

93095, 

35 

26 

562790 

968877 

593914 

406086 

36542 

93084 

34 

27 

563112 

968827 

594286 

405715 

36569 

93074 

33 

28 

29 

663433 

563755 

968777 

968728 

594656 

596027 

405344 

404973 

36596 

36623 

93063 

93052 

32 

31 

30 

664075 

968678 

595398 

404602 

36650 

93042 

30 

31 

9.564396 

9.968628 

9.595768 

10.404232 

36677 

93031 

29 

32 

564716 

968578 

O*o 

8.3 

8.3 

8.3 

8.3 

8.3 

8.3 

8.3 

8.4 
8.4 
8.4 
8.4 
8.4 
8.4 

ft A 

596138 

403862 

36704 

^3020 

28 

33 

565036 

968528 

596508 

403492 

36731 

93010 

27 

34 

35 

565356 

665676 

968479 

968429 

596878 

597247 

403122 

402753 

36758 

'36785 

92999 

92988 

26 

25 

36 

565995 

968379 

597616 

4023841 

36812 

92978 

24 

37 

666314 

968329 

597985 

402015 

36839 

92967 

23 

38 

666632 

968278 

698364 

401646 

36867 

92956 

22 

39 

566951 

968228 

598722 

401278 

36894 

92945 

21 

40 

667269 

968178 

599091 

400909 

36921 

92935 

20 

41 

9.667587 

9.968128 

9.699459 

10.400541 

369-18 

92926 

19 

42 

567904 

968078 

599827 

400173 

36975 

92913 

18 

43 

568222 

968027 

600194 

399806 

37002 

92902 

17 

44 

568539 

967977 

600562 

399438 

399071 

37029 

92892 

16 

45 

668856 

967927 

600929 

37056 92881 

16 

46 

669172 

967876 

o . 4 

8.4 

8.4 

8.4 

e A 

601296 

398704 37083 

92870 

14 

47 

569488 

• 967826 

601662 

398338 

37110 

92859 

13 

48 

569804 

967775 

602029 

3979711 

37137 

92849 

12 

49 

570120 

967725 

602395 

397605 

37164 

92838 

11 

50 

570435 

967674 

O * 

8.4 

Q A 

602761 

397239 

37191 

92827 

10 

51 

9.670751 

9.967624 

9.603127 

10.396873 

37218 

92816 

9 

52 

671066 

967573 

O . 

8.4 

8.5 

8.5 

8.6 
8.5 

8.5 

8.6 
8.6 

603493 

396507 

37245 

92805 

8 

53 

571380 

967522 

603858 

396142 

37272 

92794 

7 

54 

571695 

967471 

604223 

395777 

37299 

92784 

6 

55 

672009 

967421 

604588 

395412 

37326 

92773 

5 

56 

572323 

967370 

604953 

395047 

37353 

92762 

4 

57 

572636 

967319 

605317 

394683 

37380 

92751 

3 

68 

672950 

967268 

605682 

394318 

37407 

92740 

2 

59 

673263 

967217 

60 >046 

393954 

37434 

92729 

1 

SO 

573575 

967166 

606410 

393590| 

37461 

92718 

0 


Cosine. 


Sine. 


Cotang. 


Tang. 1 

N. cos. 

N.Sine 

/ 





68 Degrees. 







v. 

































































— —-=1 

I/Og. Sines and Tangents (22°) Natural Sines . 43 j 


/ 

Sine . 

D . 11 / 

Cosine . | D . 10 ' 

Tang . 

D . 10 " 

Cotang . N.sine 

N . 008 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 
26 
26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 
46 

46 

47 

48 
19 
60 
61 
62 
63 

54 

55 
66 
67 
58 
69 
60 

9 573575 
573888 
574200 
574512 
574824 
575136 
575447 
575758 
676069 
576379 
676689 
9.576999 
577309 
677618 
577927 
678236 
678545 
578853 
579162 
679470 
679777 
9.580085 
680392 
580699 
681005 
681312 
581618 
681924 
682229 
682535 
682840 
9.583145 
683449 
683764 
684058 
684361 
684665 
684968 
685272 
585574 
585877 
9.586179 
586482 
686783 
687085 
587386 
687688 
587989 
688289 
688690 
688890 
9.689190 
689489 
689789 
690088 
£90387 
590686 
690984 
691282 
691580 
691878 

52.1 
52.0 
52.0 

51.9 

61.9 

51.9 
51.8 

51.8 

51.7 

61.7 
51.6 

51.6 

61.6 
61.6 

51.6 

51.4 

61.4 

51.3 

61.3 

51.3 

51.2 

51.2 
61.1 
51.1 

51.1 
51.0 
51.0 

50.9 
50.9 
50.9 

60.8 

50.8 

60.7 

60.7 

50.6 

60.6 
60.6 

60.5 

50.6 

50.4 
50.4 

50.3 
50.3 
50.3 

50.2 
50.2 
60.1 
50.1 
50.1 
50.0 
50.0 

49.9 
49.9 
49.9 

49.8 
49.8 

49.7 
49.7 
49.7 
49.6 

9.967166 
967115 
967064 
967013 
966961 
966910 
966859 
966808 
966756 
966705 
966653 
9.966602 
966550 
966499 
966447 
966395 
966344 
966292 
966240 
966188 
966136 
9.966085 
966033 
965981 
965928 
965876 
965824 
965772 
966720 
965668 
965615 
9.965563 
965511 
965458 
965406 
966353 
965301 
965248 
965195 
965143 
965090 
9.965037 
964984 
964931 
964879 
964826 
964773 
964719 
964666 
964613 
964560 
9.964507 
964454 
964400 
964347 
964294 
964240 
964187 
964133 
964080 
964026 

8.5 

8.6 
8.5 
8.5 
8.5 

8.5 

8.6 

8.5 

8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.6 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 
8.7 

8.7 

8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.8 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 
8.9 

9.606410 
6067 i 3 
60/137 
607500 
60/863 
608225 
608588 
608950 
609312 
609674 
610036 
9.610397 
610759 
611120 
611480 
611841 
612201 
612561 
612921 
613281 
613641 
9.614000 
614359 
614718 
615077 
615435 
615793 
616151 
616509 
616867 
617224 
9.617582 
617939 
618295 
618652 
619008 
619364 
619721 
620076 
620432 
620787 
9.621142 
621497 
621852 
622207 
622561 
622916 
623269 
623623 
623976 
624330 
9.624683 
625036 
625388 
625741 
626093 
626445 
626797 
627149 
627601 
627852 

60.6 

60.6 

60.6 

60.5 
60.4 
60.4 

60.4 
60.3 
60.3 

60.3 
60.2 
60.2 
60.2 
60.1 
60.1 
60.1 
60.0 
60.0 
60.0 
59.9 
59.9 
69.8 

69.8 

59.8 
59.7 
59.7 

59.7 

59.6 
59.6 
59.6 

59.5 

59.6 

59.5 

59.4 
59.4 
59.4 

69.3 

59.3 
59.3 
59.2 
59.2 
59.2 

59.1 

69.1 
59.0 
59.0 
59.0 

58.9 
58.9 
58.9 

58.8 
58.8 
58.8 

58.7 

68.7 

58.7 

58.6 
58.6 
58.6 
58.6 

10.393590 
393227 
392863 
392500 
392137 
391776 
391412 
391050 
390688 
390326 
389964 
10.389603 
389241 
388880 
388520 
388159 
387799 
387439 
387079 
386719 
386359 
10-386000 
385641 
385282 
384923 
384566 
384207 
383849 
383491 
383133 
382776 
10-382418 
382061 
381705 
381348 i 
380992 ! 
380636 [ 
380279 
379924 
379568 
379213 
10-378858 
378503 
378148 
377793 
377439 
377085 
376731 | 
376377 
376024 
375670 
10-375317 
374964 
374612 
374259 
373907 
373655 
373203 
372851 
372499 
372148 

37401 

137488 
37515 
:37642 
37569 
37595 
37622 
;37649 
i 37676 
37703 
37730 
37757 
37784 
37811 
37838 
37866 
37892 
37919 
37946 
37973 
37999 
38026 
38053 
38080 
38107 
38134 
38161 
38188 
38215 
38241 
38268 
38295 
38322 
38349 
38376 
38403 
38430 
38456 
38483 
38510 
38537 
38564 
38591 
38617 
38644 
38671 
38698 
38725 
38762 
38778 
38805 
38832 
38859 
38886 
38912 
38939 
38966 
38993 
39020 
39046 
390/3 

92718 

92/07 

92697 

92686 

92675 
92664 
92653 
92642 

92631 
92620 
92609 
92598 
92587 

92676 
92565 
92554 
92543 

92632 
92521 
92510 
92499 
92488 
92477 
92466 
92455 
92444 
92432 
92421 
92410 
92399 
92388 
92377 
92366 
92355 
92343 
92332 
92321 
92310 
92299 
92287 
92276 
92265 
92264 
92243 
92231 
92220 
92209 
92198 
92186 
92175 
92164 
92152 
92141 
92130 
92119 
92107 
92096 
92085 
92073 
92062 
92050 

60 

59 

58 

5 ? 

66 

55 

54 

53 

52 

61 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

26 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

t 

Cosine . 


Sine . 


Cotang . 


Tang . 

N . cos . 

N . sine . 


67 Degrees . 


19 















































































44 Log. Sines and Tangents. (23°) Natural Sines. TABLE II. 


/ 

Sine. 

D lu' 

Cosine. 

D. 10 

Tang. 

D. to' 

j Cotang. 

N. sine. 

N. cos. 


0 

1 

2 

3 

4 

! & 

1 6 

7 

8 
9 

10 

11 

12 

13 

14 
16 
16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 

52 

53 

54 

65 

66 
67 
58 
69 
60 

9.591878 
692176 
5924?3 
692770 
693037 
593363 
593659 
593955 
694251 
694547 
594842 
9.595137 
595432 
695727 
596021 
696315 
696609 
596903 
697196 
597490 
597783 
9.598075 
598368 
598660 
598952 
599244 
599536 
699827 
600118 
600409 
600700 
9 600990 
601280 
801570 
601860 
602150 
602439 
602728 
603017 
603305 
603594 
b 603882 
604170 
604457 
604745 
805032 
805319 
605606 
605892 
608179 
606465 
S 606761 
607036 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 

49.6 
49.5 
49.5 
49.5 
49.4 

49.4 
49.3 

49.3 

49.5 
49.2 

49.2 
49.1 
49.1 

49.1 
49.0 
49.0 
48.9 
48.9 
48.9 
48.8 

48.8 

48.7 
48.7 

48.7 

48.6 
48.6 
48.6 
48.6 

48.6 

48.4 
48.4 
48.4 

48.3 

48.3 

48.2 
48.2 

48.2 
48.1 
48.1 
48.1 
48.0 
48.0 

47.9 
47.9 
47.9 
47 8 
47-8 

47.8 

47.7 
47-7 
476 
47-6 
47-6 
47-6 
47-6 
47-4 

47.4 
47.4 

47.3 
47.3 

9.964026 

963972 

963919 

963865 

963811 

963757 

963704 

963650 

963596 

963542 

963488 

9.963434 

963379 

963325 

963271 

963217 

963163 

963108 

963054 

962999 

962945 

9.962890 

962836 

962781 

962727 

962672 

962617 

962562 

962508 

962453 

962398 

9.962343 

962288 

962233 

962178 

962123 

962067 

962012 

961957 

961902 

961846 

9.961791 

961735 

961680 

961624 

961569 

961513 

961458 

961402 

961346 

961290 

9.961235 

961179 

961123 

961067 

961011 

960955 

960899 

960843 

960786 

960730 

8.9 

8.9 

8.9 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.0 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9.1 

9 2 

9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 
9.2 

9.2 

9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 
9.3 

9.3 

9.4 
9.4 

9.627852 
628203 
628554 
628905 
629255 
629606 
629956 
630306 
630656 
631005 
631365 
9.631704 
632053 
632401 
632750 
633098 
633447 
633795 
634143 
634490 
634838 
9.635185 
635532 
635879 
636226 
636572 
636919 
637265 
637611 
637956 
638302 
9.638647 
638992 
639337 
639682 
640027 
640371 
640716 
641060 
641404 
641747 
9.642091 
642434 
642777 
643120 
6434133 
643806 
644148 
644490 
644832 
645174 
9.645516 
645857 
646199 
646540 
646881 
647222 
• 647562 
647903 
648243 
648583 

58.5 

68.6 

58.5 
58.4 

58.4 

68.3 

58.3 

58.3 

68.3 
58.2 
68.2 

58.2 
68.1 
58.1 

58.1 
68.0 
68.0 
68.0 
57.9 
57.9 
57.9 
57.8 
57.8 

57.8 

67.7 

57.7 

67.7 

67.7 

67.6 

57.6 

57.6 

67.6 

57.5 
57.5 

67.4 
67.4 
67.4 

57.3 

67.3 
67.3 

57.2 

57.2 

67.2 
67.2 

67.1 

57.1 

67.1 
67.0 
57.0 
67.0 

66.9 

66.9 

56.9 

66.9 

66.8 
66.8 
66.8 

66.7 
66.7 
66.7 

10.372148 

371797 

371446 

371096 

370746 

370394 

370044 

369694 

369344 

3689y5 

368645 

10.368296 

367947 

367599 

367250 

366902 

366553 

366205 

365857 

365510 

365162 

10.364815 

364468 

364121 

363774 

363428 

363081 

362735 

362389 

362044 

361698 

10.361353 

361008 

360663 

360318 

359973 

359629 

359284 

368940 

358596 

358253 

10.367909 

367666 

367223 

356880 

356637 

356194 

356852 

355510 

355168 

354826 

10.354484 

354143 

363801 

363460 

353119 

352778 

352438 

352097 

351767 

361417 

39073 
39100 
39127 
39153 
39180 
39207 
!39234 
!39260 
39287 
39314 
39341 
39367 
39394 
39421 
!39448 
39474 
39501 
39528 
39555 
39581 
39608 
39636 
39661 
39688 
139715 
39741 
39768 
39795 
39822 
39848 
39876 
39902 
39928 
39955 
39982 
40008 
40035 
40062 
40088 
40115 
40141 
40168 
40195 
40221 
40248 
40275 
40301 
40328 
40356 
40381 
40408 
40434 
40461 
40488 
40514 
40541 
40567 
40594 
40621 
40647 
40674 

92050 

92039 

92028 

92016 

92005 

91994 

91982 

91971 

91959 

91948 

91936 

91925 

91914 

91902 

91891 

91879 

91868 

91856 

91845 

91833 

91822 

91810 

91799 

91787 

91775 

91764 

91762 

91741 

91729 

91718 

91706 

91694 

91683 

91671 

91660 

91648 

91636 

91625 

91613 

91601 

91590 

91578 

91566 

91565 

91543 

91531 

91519 

91508 

91496 

91484 

91472 

91461 

91449 

91437 

91425 

91414 

91402 

91390 

91378 

91366 

91355 

60 

69 

58 

57 

56 

65 

54 

53 

52 

61 

50 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

26 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

t 


66 Degrees. 






























































TABUS II. Log. Sines and Tangents. (24°) Natural Sines. 


45 


0 
1 

2 

3 

4 
6 
6 

7 

8 
9 

10 
11 
12 

13 

14 
16 
16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 

51 

52 

63 

64 
55 
66 
67 

58 

59 

60 


Sine. 


9.609313 
609507 
6098 ^0 
610164 
610147 
610729 
611012 
611294 
611576 
611858 
612140 
9.61242A 
612702 
612983 
613264 
613545 
613825 
614105 
61438 
614666 
61494 
9.615223 
615502 
615781 
616060 
616338 
616616 
616894 
617172 
617450 
617727 
9.618004 
618281 
618558 
618834 
619110 
619386 
619662 
619938 
620213 
620488 
.620763 
621038 
621313 
621587 
621861 
622135 
622409 
622682 
622956 
623229 
9.623512 
623774 
624047 
624319 
624591 
624863 
625136 
625406 
625677 
625948 


D. 10 


Cosine. 


47. 
47. 
47. 
47. 
47. 
47. 
47. 
47. 
47. 
46. 
46. 
46. 
46. 
46.8 
46.7 
46.7 
46.7 
46.6 
46.6 
46.6 
46.5 

46.5 

46.6 
46 
46 
46 
46.3 
46.3 
46.2 
46.2 
46.2 
46.1 
46.1 
46.1 
46.0 
46.0 
46.0 
45.9 
45.9 
45.9 
45.8 
45.8 

45.7 
45.7 
45.7 
45.6 
45.6 
45.6 
45.6 
45.6 
45.6 
45.4 
45.4 
45.4 
45.3 

45.3 

46.3 
45.2 
45.2 
45.2 


Cosine. 


950730 
960574 
960618 
960551 
960505 
960448 
960392 
960335 
960279 
960222 
960165 
9.950109 
960052 
959995 
959938 
959882 
959825 
959768 
959711 
959654 
959596 
9.959539 
959482 
959425 
959368 
959310 
959263 
959195 
959138 
959081 j 
959023 
9.958965 
958908 
958850 
958792 
958734 
958377 
958619 
958561 
958503 
958445 
.958387 
958329 
958271 
958213 
958154 
958096 
958038 
957979 
957921 
957863 
9.967804 
957746 
957687 
957628 
957570 
957511 
957452 
957393 
967335 
957276 


D. lu' 


Sine. 


9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.4 

9.5 

9.6 
9.6 
9.6 
9.6 

9.5 

9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.5 

9.5 

9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 
9.6 

9.6 

9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 
9.7 

9.7 

9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 
9.8 


Tan’. 


662709 
663042 
663375 
663707 
664039 
664371 
664703 
665035 
665366 
9.665697 
666029 
666360 
666691 
667021 
667352 
667682 
668013 
668343 
668672 


Cotang. 


D to 

’ Cotang. 

1 N. sint 

>. N. cot 

1. 

13 _ . 
>3 

■*o OO . v 

>3 Kli t. 

n J®-® 
12 j* 

0 it i 
9 SI 

7 SI 

/j oo . ^ 

4 66.4 
o 56.3 
t 66.3 
r, 66.3 

2 56.3 

o 56.2 

n 56 2 
9 56.2 

■» S 

1 S 

8 S' 

4 56.1 

9 S'n 
6 S' 

t 55.9 
4 65.9 
t 55.9 
\ 65.9 

* r* t? Q 

3 OO.o 

'J f r o 

t 55 .8 
J 65.8 
\ 65.8 

J 55.7 
> 65.7 
: 55.7 
! 55.7 
55.6 

55.6 

65.6 

65.6 

55.6 
55.5 
65.4 

10.351417 

351077 

4087' 
4070( 

1 9136E 
191343 

60 

59 

350737 

4072' 

91331 

68 

350398 

40767 

91319 

57 

350058 

4078( 

191307 

66 

349719 

4080t 

>.91296 

56 

349380 

40837 

91283 

64 

349041 

40861 

191272 

53 

348703 

4088t 

>91260 

62 

348364 

348026 

40913 

40931) 

91248 

•91236 

61 

50 

10.347688 

4096(1 

91224 

49 

347350 

40992 

91212 

48 

347012 

41019 

91200 

47 

346674 

41045 

91188 

46 

346337 

41072 

91176 

45 

346000 

41098 

01164 

44 

345663 

41125 

91152 

43 

345326 

41151 

91140 

42 

344989 

41178 

91128 

41 

344652 

41204 

91116 

40 

10.344316 

41231 

91104 

39 

343980 

41257 

91092 

38 

343644 

41284 

91080 

37 

343308 

41310 

91068 

36 

342972 

41337 

91066 

35 

342636 

41363 

91044 

34 

342301 

41390 

91032 

33 

341966 

4141691020 

32 

341631 

41443 91008 

31 

341296 

41469 90996 

30 

10.340961 

41496 90984 

29 

340627 

41522 90972 

28 

340292 

41549 90960 

27 

339958 

41576 90948 

26 

339624 

41602 

90936 

25 

339290 

41628 

90924. 

24 

338957 

41655 

90911 

23 

338623 

41681 

90899 

22 

338290 

41707 

90887 

21 

337957 

41734 

90875 

20 

10.337624 

41760 

90863 

19 

337291 

41787 

90861 

18 

336958 

41813! 

90839 

17 

55.4 

55.4 

55.4 

65.3 

65.3 

55.3 
55.3 
65.2 
65.2 
65.2 
65.1 

65.1 

55.1 

65.1 
55.0 
65.0 
55.0 

336626 

41840190826 

16 

336293 

41866! 

90814 

15 

335961 

41892 

908 02 

14 

335629 

41919! 

90790 

13 

335297 

41945! 

90778 

12 

334965 

41972! 

90766 

11 

334634 

41998! 

90753 

10 

0.334303 

42024 ! 

90741 

9 

333971 

42051 ! 

907 29 

8 

333620 

42077 ! 

907 17 

7 

335309 i 

42104 9 

0704 

6 

332979 

42150 9 

0692 

5 

332648 ; 

42lo»> 9 

ia, 8 u 

4 I 

332318 

42183 9 

0668 

3 

331987 

42209 9 

0656 

2 

331657 

422369 

0643 

1 

331328 

42262 9 

Oool 

0 


Tang. 

N. cos. J 

«.sine. 

r 


65 Degrees. 





























































































46 


Log. Sines and Tangents. (2a 1 ) Natural Sines. TABLE II. 


Sine. 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 
48 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9. 


D. 10' 


9. 


625948 
626219 
626490 
6267o0 
627030 
627300 
627570 
627840 
628109 
628378 
628647 
628916 
629185 
629453 
629721 
629989 
630257 
630524 
630792 
631C<^9 
631326 
631593 
631859 
632125 
632392 
632658 
632923 
633189 
633454 
633719 
633984 
634249 
634514 
634778 
635042 
635303 
635570 
635834 
636097 
636360 
636623 
.636886 
63714S 
637411 
637673 
637936 
638197 
638458 
638720 
638981 
639242 
.639503 
639764 
640024 
640284 
640544 
640804 
641064 
641324 
641584 
641842 


45.1 

45.1 

45.1 
45.0 
45.0 
45.0 
44.9 
44.9 
44.9 
44.8 

44.8 
44.7 
44.7 

44.7 
44.6 
44.6 
44.6 

44.6 
44.5 
44.5 

44.5 
44.4 
44.4 

44.4 
44.3 
44.3 

44.3 

44.2 
44.2 

44.2 
44.1 

44.1 
44.0 
44.0 
44.0 

43.9 
43.9 
43.9 

43.8 
43.8 
43.8 

43.7 
43.7 
43.7 
43.7 

43.6 
43.6 
43.6 

43.5 
43.5 
43.5 

43.4 
43.4 
43.4 

43.3 
43.3 
43.3 

43.2 
43.2 
43.2 


9. 


Cosine. 


D. 10' 


Cosine. 


957276 

957217 

957158 

957099 

957040 

956981 

956921 

956862 

955803 

956744 

956684 

956625 

956566 

956506 

956447 

955387 

956327 

955268 

956208 

956148 

95608y 

956029 

955969 

955909 

955849 

955789 

955729 

955669 

955609 

955548 

955488 

,955428 

955368 

955307 

955247 

955186 

955126 

955065 

955005 

954944 

951883 

.954823 

954762 

954701 

954640 

954579 

954518 

954457 

954396 

954335 

954274 

.954213 

954152 

954090 

954029 

953968 

953906 

953845 

953783 

953722 

953660 


9.8 

9.8 

9.8 

9.8 

9.8 

9.8 

9.9 
9.9 
9.9 
9.9 
9.9 
9.9 


9, 

9 

9 


Sine. 


9.9 

9.9 

9.9 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.0 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.1 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.2 

10.3 


Tang. D. 10" 


9.668673 
669002 
669332 
669661 
669991 
670320 
670649 
670977 
671306 
671634 
671963 
672291 
672619 
672947 
673274 
673602 
673929 
674257 
674584 
674910 
675237 
675564 
675890 
676216 
676543 
676869 
677194 
677520 
677846 
678171 
678496 
9.678821 
679146 
679471 
679795 
680120 
680444 
680768 
681092 
681416 
681740 
9.682063 
682387 
682710 
683033 
683356 
683679 
684001 
684324 
684646 
684968 
9.685290 
685612 
685934 
686255 
686577 
686898 
687219 
687540 
687861 
688182 


55.0 

54.9 

54.9 

54.9 

64.8 

54.8 
54.8 

54.8 

54.7 

64.7 

64.7 

54.7 

64.6 

54.6 
54.6 

54.6 

64.5 

54.5 

54.5 
54.4 
54.4 
54.4 

54.4 
54.3 
54.3 
54.3 

54.3 
54.2 
54.2 
54.2 
54.2 

64.1 

54.1 

64.1 

54.1 
54.0 
54.0 
54.0 
54.0 

53.9 

53.9 

63.9 
63.9 

53.8 

53.8 

63.8 

53.8 

53.7 

63.7 

53.7 
53.7 

53.6 
53.6 
53.6 
53.6 

53.5 
53.5 
53.5 
53.5 

53.4 


Cotang. 


N .sine. N. cos 


Co tang. 


10.331327 
330998 
330668 
330339 
330009 
329680 
329351 
329023 
328694 
328366 
328037 
10.327709 
327381 
327053 
326726 
326398 
326071 
325743 
325416 
325090 
324763 
10.324436 
324110 
323784 
323457 
323131 
322806 
322480 
322154 
321829 
321504 
10.321179 
320854 
320529 
320205 
319880 
319556 
319232 
318908 
318584 
318260 
10.317937 
317613 
317290 
316967 
316644 
316321 
315999 
315676 
315354 
315032 
314710 
314388 
314066 
313745 
313423 
313102 
312781 
312460 
312139 
311818 


10 . 


42262 
42288 
42315 
42341 
42367 
42394 
42420 
42446 
42473 
42499 
42525 
42552 
42578 
42604 
42631 
42657 
42683 
42709 
42736 
42762 
42788 
42815 
42841 
42867 
42894 
42920 
42946 
42972 
|42999 
43025 
43051 
43077 
43104 
43130 
'43156 
43182 
43209 
43235 
43261 
■43287 
I 43313 
43340 
:43366 
43392 
43418 
43445 
143471 
I 43497 
43528 
43549 
! 143575 
|43602 
I! 43628 


90631 

90613 

90606 

90594 

90582 

90569 

90557 

90545 

90532 

90520 

90507 

90495 

90483 

90470 

90158 

90446 

90433 

90421 

90408 

90396 

90383 

90371 

90358 

90346 

90334 

90321 

90309 

90296 

90284 

90271 

90259 

90246 

90233 

90221 

90208 

90196 

90183 

90171 

90158 

90146 

90133 

90120 

90108 

90095 

90082 

90070 

90057 

90045 

90032 

90019 

90007 

89994 

89981 

89968 


Tang. 


43654 
43680 89955 
43706 
43738 
43759 
43785 
43811 
43837 


N. eo«. N .nii 


89943 

89930 

89918 

89905 

89892 

89879 


60 

69 

68 

57 

56 

55 

54 

63 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


64 Degrees. 

























































- - —- ' 

Lo£. Sines aud Tangents. (26°) Natural Sines. 47 


/ 

j Sine. 

D. 10' 

Cosine. 

D. 10' 

Tang. 

D. 10' 

'j Cotaug. 

N. sine 

. N. cos 


0 

1 

2 

3 

4 
B 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 

52 

53 

54 

55 
66 
67 

58 

59 

60 

9.641842 

642101 

642360 

642618 

642877 

643135 

643393 

643650 

643908 

644165 

644-123 

9.644550 

644936 

645193 

645450 

645706 

645962 

646218 

646474 

646729 

646984 

9.647240 

647494 

647749 

648004 

648258 

645612 

648766 

649020 

649274 

649627 

9.649781 

650J34 

650287 

650539 

650792 

651044 

651297 

651549 

651800 

652052 

9.652304 

652555 

652806 

653057 

653308 

653558 

653808 

654059 

654309 

654558 

9.654808 

655058 

655307 

655556 

655805 

656054 

656302 

656551 

656799 

657047 

43. 1 
43.1 

43.1 
43.0 
43.0 
43.0 
43.0 
42.9 
42.9 
42.9 
42.8 
42.8 

42.8 
42.7 
42.7 
42.7 
42.6 
42.6 
42.6 
42.5 
42.5 
42.5 
42.4 
42.4 
42.4 
42.4 
42.3 
42.3 
42.3 

42.2 
42.2 
42.2 
42.2 
42.1 
42.1 
42.1 
42.0 
42.0 
42.0 

41.9 
41.9 
41.9 
41.8 
41.8 
41.8 
41.8 
41.7 
41.7 
41.7 
41.6 
41.6 
41.6 
41.6 
41.6 
41.5 
41.5 
41.4 
41.4 
41.4 
41.3 

9.953660 

953599 

953537 

953475 

953413 

953352 

953290 

953228 

953166 

953104 

953042 

9.952980 

952918 

952855 

952793 

952731 

952669 

952606 

952544 

9524S1 

952419 

9.952356 

952264 

952231 

952168 

952106 

952043 

951980 

951917 

951854 

951791 

9.951728 

951665 

951602 

951539 

951476 

951412 

951349 

951286 

951222 

951159 

9 951096 
951032 
950968 
950905 
950841 
950778 
950714 
950650 
950586 
950522 

9 950458 
950394 
950330 
950366 
950202 
96013S 
950074 
950010 
949946 
.949881 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.3 

10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 
10.4 

10.4 

10.5 

10.5 

10.6 
10.6 
10.5 
10.5 
10.5 
10.5 
10.5 
10.5 
10.5 

10.5 

10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.6 
10.7 
10.7 
10.7 
10.7 
10.7 
10.7 
10.7 
10.7 
10.7 
10.7 

9.688182 
688502 
688823 
689143 
689463 
689783 
690103 
690-123 
690742 
691062 
691381 
9.691700 
692019 
692338 
692656 
692975 
693293 
693612 
693930 
694248 
694666 
9.694883 
695201 
695618 
695836 
696153 
696470 
696787 
697103 
697420 
697736 
9.698053 
698369 
698685 
699001 
699316 
699632 
699947 
700263 
700578 
700893 
9.701208 
701623 
701837 
702162 
702466 
702780 
703095 
703409 
703723 
704036 

9 704360 
704663 
704977 
705290 
705603 
705916 
706228 
706541 
706864 
707166 

53.4 

53.4 

63.4 

63.3 

53.3 

63.3 

53.3 
53.3 
53.2 

53.2 

63.2 

53.1 

63.1 

63.1 

53.1 

63.1 
63.0 
53.0 
53.0 
53.0 
52.9 
52.9 
52.9 
52.9 
52.9 

52.8 

62.8 
62.8 
52.8 
62.7 
62.7 
62.7 
62.7 
62.6 
62.6 
62.6 
62.6 
52.6 

52.6 

62.6 
62.6 
62.4 
62.4 
62.4 
62.4 
62.4 
62.3 

62.3 

52.3 
52.3 
62.2 
62.2 

52.2 

62.2 
62.2 
52.1 

52.1 

62.1 
52.1 
52.1 

10.311818 
311498 
311177 
310857 
310537 
310217 
309897 
309577 
309258 
308938 
308619 
10.308300 
307981 
307662 
307344 
307025 
306707 
306388 
306070 
305752 
305434 
10-305117 
304799 
304482 
304164 
303847 
303630 
303213 
302897 
302580 
302264 
10-301947 
301631 
301315 
300999 
300684 
300368 
300058 
299737! 
299422i 
299107 
0-298792 
298477 
298163 
297848 
297634 
297220 
296905 
296591 
296277 
295964 
10-295650 
295337 
295023 
294710 
294397 
294084 
293772 
293469 
293146 
292834 

114383' 
1 43867 
43881 
43916 
43942 
I 43968 
1 4399-1 
| 44020 
44040 
| 44072 
44098 
44124 
:44151 
44177 
!44203 
44229 
44255 
44281 
44307 
44333 
44359 
44386 
44411 
44437 
44464 
44490 
44516 
44542 
44568 
44594 
44620 
44646 
44672 
44698 
44724 
44750 
44776 
44802 
44828 
44854 
44880 
44906 
44932 
44958 
44984 
45010 
45036 
45062 
450881 
451141 
451401 
451661 
451921 
452181 
45243 
45269 
45295 
45321 
45347 
453731 
45599 

89879 

89867 

89854 

89841 

89828 

89816 

89803 

89790 

89777 

89764 

89752 

89739 

89726 

89713 

89700 

89687 

89674 

89662 

89649 

89636 

89623 

89610 

89697 

89584 

89571 

89558 

89645 

89532 

89519 

89506 

89493 

89480 

89467 

89454 

89441 

89428 

89415 

89402 

89389 

89376 

89363 

89350 

89337 

89324 

89311 

89298 

89285 

89272 

89269 

89245 

89232 

89219 

89206 

39193 

89180 

89167 

89153 

89140 

89127 

89114 

89101 

60 

59 
58 
57 
56 
55 
64 
63 
62 
61 

60 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
26 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 | 


Cosine. 


Sine. 


Cctaii^ | 


Tang 

N. cos. 

tf.sine. 

' | 


63 Degrees. 











































































48 Log. Sines and Tangents. (27°) Natural Sines. TABLE II. 


/ 

Sini. 1 

J. 1 o' 

(josme. 

0. j/ 

1 > lJ b. 

D. io 

UOtang. 

S. sine. 

N. cos. 


0 

1 

2 

3 

4 

B 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

). 657047 
657295 
657542 
657790 
658037 
658284 
658531 
658778 
659025 
659271 
659517 
9.659763 
660009 
660255 
660501 
660746 
660991 
661236 
661481 
661726 
661970 
9.662214 
662459 
662703 
662946 
663190 
663433 
663677 
663920 
664163 
664406 
9.664648 
664891 
665133 
665375 
665617 
665859 
666100 
666342 
666583 
666824 
9.667065 
667305 
667546 
667786 
668027 
668267 
668506 
668746 
668986 
669225 
9.669464 
669703 
669942 
670181 
670419 
670658 
670896 
671134 
671372 
671609 

41.3 

41.3 
41.2 
41.2 
41.2 

41.2 
41.1 
41.1 

41.1 
41.0 
41.0 
41.0 
40.9 
40.9 
40.9 
40.9 
40.8 
40.8 

40.8 
40.7 
40.7 
40.7 

40.7 
40.6 
40.6 

40.6 
40.5 
40.5 
40.5 

40.5 

40.4 
40.4 
40.4 

40.3 
40.3 
40.3 

40.2 
40.2 
40.2 
40.2 
40.1 
40.1 
40.1 
40.1 
40.0 
40 0 
40.0 

39.9 
39.9 
39.9 
39.9 

39.8 
39.8 
39.8 

39.7 
39.7 
39.7 
39.7 
39 6 

39.6 

).949881 
949816 
949752 
949688 
949623 
949558 
949494 
949429 
949364 
949300 
949235 
9.949170 
949105 
949040 
948975 
948910 
948845 
948780 
948715 
948650 
948584 
9.948519 
948454 
948388 
948323 
948257 
948192 
948126 
948060 
947995 
947929 
9.947863 
947797 
947731 
947665 
947600 
947533 
947467 
947401 
947335 
947269 
9.947203 
947136 
947070 
947004 
946937 
946871 
946804 
946738 
946671 
946604 
9.946538 
946471 
946404 
946337 
946270 
946203 
946136 
946069 
946002 
945935 

10.7 

10.7 

10.7 

10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.8 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
10.9 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.0 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.1 
11.2 
11.2 
11.2 
11.2 
11.2 

3.70 1 166 
707478 
707790 
708102 
708414 
708726 
709037 
709349 
709660 
709971 
710282 
9.710593 
710904 
711215 
711525 
711836 
712146 
712453 
712766 
713076 
713386 
9.713696 
714005 
714314 
714624 
714933 
715242 
715651 
715860 
716168 
716477 
9.716785 
717093 
717401 
717709 
718017 
718325 
718633 
718940 
719248 
719555 
9.719862 
720169 
720476 
720783 
721089 
721396 
721702 
722009 
722315 
722621 
9.722927 
723232 
723538 
723844 
724149 
724454 
724759 
725065 
725369 
725674 

52.0 

52.0 

52.0 

52.0 

51.9 

51.9 

51.9 

51.9 

51.9 

51.8 

51.8 

51.8 

51.8 

51.8 
51.7 
51.7 
51.7 

51.7 
51.6 
51.6 
51.6 
51.6 
51.6 
51.5 
51.5 
51.5 
51.5 

51.4 

61.4 

51.4 

51.4 

61.4 

51.3 

61.3 

51.3 
51.3 
51.3 
61.2 

51.2 

61.2 
61.2 
51.2 
61.1 
51.1 
51.1 
51.1 
51.1 
51.0 
51.0 
51.0 
51.0 
61.0 

50.9 

60.9 

50.9 
50.9 
50.9 

50.8 
50.8 
50.8 

i 

10.292834!' 
292522} 
292210 
291898 
291586! 
291274, 
290963 
290661 
290340 
290029 
289718 
10.289407 
289096 
288785 
288476 
288164 
28i854 
287644 
287234 
286924 
286614 
10.286304 
285995 
286686 
285376 
285067 
284768 
28444.9 
284140 
283832 
283523 
10.283216 
282907 
282599 
282291 
281983 
281675 
281367 
281060 
280752 
280445 
10.280138 
279831 
279624 
279217 
278911 
278604 
278298 
277991 
277685 
277379 
It 277073 
276768 
276462 
276156 
275851 
276546 
275241 
274935 
274631 
274326 

45399 
45425 
45451 
45477 
46503 
45529 
45554 
45580 
45606 
45632 
456581 
45684 
45710 
45736 
45762 
45787 
45813 
45839 
45865! 
45891 
45917 
45942 
45968 
45994 
46020 
46046 
46072 
46097 
46123 
46149 
46175 
46201 
46226 
46252 
46278 
46304 
46330 
46365 
46381 
46407 
46433 
46458 
46484 
,46510 
146536 
46661 
146587 
146613 
1 46639 
46664 
46696 
46716 
46742 
46767 
46793 
46819 
46844 
46871 
46896 
46921 
4694' 

89101 

89087 

89074 

89061 

89048 

89036 

890-21 

89008 

88995 

88981 

88968 

88955 

88942 

88928 

88915 

88902 

88888 

88875 

88862 

88848 

88835 

88822 

88808 

88795 

88782 

88768 

88755 

88741 

88728 

88715 

88701 

88688 

88674 

88661 

88647 

88634 

88620 

88607 

38593 

88680 

88566 

88553 

88539 

88526 

88512 

88499 

88485 

88472 

88458 

88445 

88431 

88417 

88404 

88390 

88377 

88363 

88349 

188336 

88322 

88308 

88295 

60 

69 

58 

57 

66 

56 

54 

53 

52 

61 

60 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

36 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


| Cotang. | 

Tang. 

| N. cos 

.. N.sino 

/ 


62 Degrees. 










































































— V 

TABLE II. Log. Sines and Tangents. (28°) Natural Sines. 49 

1 ° 

1 2 

1 3 

4 

6 

: 6 

7 

8 
9 

1 10 

1 11 
12 

13 

14 

15 

16 

17 

18 

19 

20 
21 

I 22 

I 23 
24 

25 

26 
27 

1 28 

1 29 

1 30 

1 31 

32 

1 33 

1 34 
35 

36 

1 37 

II 38 

II 39 

1 40 

1 41 

I 42 

I 43 

1 44 

1 45 

1 46 

1 47 
48 

1 49 

1 60 

1 51 

I 52 

1 63 

1 64 

55 

56 

57 

1 58 

59 

1 60 

Z3U19. 

9.671699 
671847 
672034 
672321 
672558 
672795 
673032 
673268 
673505 
673741 
673977 
9.674213 
674448 
674684 
674919 
675155 
675390 
675624 
675859 
676094 
676328 
9.676562 
676796 
677030 
677264 
677498 
677731 
677964 
678197 
678430 
678663 
9.678895 
679128 
679360 
679592 
679824 
680056 
680288 
680519 
680750 
680982 
9.681213 
681443 
681674 
681905 
682135 
682365 
682595 
682825 
683055 
683284 
1.683514 
683743 
683972 
684201 
684430 
684658 
684887 
685115 
685343 
685571 

D. 10 

39.6 
39.5 
39.5 
39.5 

39.5 
39.4 
39.4 
39.4 

39.4 
39.3 
39.3 
39.3 
39.2 
39.2 
39.2 
39.2 
39.1 
39.1 
39.1 
39.1 
39.0 
39.0 
39.0 
39.0 
38.9 
38.9 
38.9 
38.8 
38.8 
38.8 
38.8 

38.7 
38.7 
38.7 
38.7 

38.6 
38.6 
38.6 

38.5 
38.5 
38.5 
38.5 
38.4 
38.4 
38.4 
38.4 
38.3 
38.3 
38.3 
38.3 
38.2 
38.2 
38.2 
38.2 
38.1 
38.1 
38.1 
38.0 
38.0 
38.0 

1 Cosine. 

D. 10' 

' Tang. 

D. 10 

" Cotang. N. sine.lN. cos 

. 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 1 

8 

7 

6 

6 

4 

3 

2 

1 

0 

9.945935 

945868 

945800 

945733 

945666 

945598 

945531 

945464 

945396 

945328 

945261 

9.945193 

945125 

945058 

944990 

944922 

944854 

944786 

944718 

944650 

£44582 

9.944514 

944446 

944377 

944309 

944241 

944172 

944104 

944036 

943967 

943899 

9.943830 

943761 

943693 

943624 

943555 

943486 

943417 

943348 

943279 

943210 

9.943141 

943072 

943003 

942934 

942864 

942795 

942726 

942656 

942587 

942517 

9.942448 

942378 

942308 

942239 

942169 

942099 

942029 

941959 

941889 

941819 

11.2 

11.2 

11.2 

11.2 

11.2 

11.2 

11.2 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.3 

11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 
11.4 

11.4 

11.5 

11.5 

11.6 
11.5 
11.5 
11.5 

11.5 

11.6 
11.6 
11.6 
11.6 
11.6 

11.5 

11.6 
11.6 
11.6 
11.61 
11.6 , 
11.6 
11.6 
11.6 
11.6 
11.6 
11.6 
11.6 
11.6 
11.7 

9.725674 

725979 

726284 

726588 

726892 

727197 

727501 

727805 

728109 

728412 

728716 

9.729020 

729323 

729626 

729929 

730233 

730535 

730838 

731141 

731444 

731746 

9.732048 

732351 

732653 

732965 

733257 

733558 

733860 

734162 

734463 

734764 

9.735066 

735367 

735668 

735969 

736269 

736570 

736871 

737171 

737471 

737771 

9.738071 

738371 

738671 

738971 

739271 

739570 

739870 

740169 

740468 

740767 

1.741066 

741366 

741664 

741962 

742261 

742559 

742858 

743156 

743454 

743752 

50.8 

60.8 
50 7 
50.7 
50.7 

50.7 

60.7 
50-6 
60.6 
50.6 
50-6 
50.6 
50.5 
50-5 
50 5 
50.5 
50.5 
50-4 
50.4 

50.4 

60.4 
60.4 
50-3 
50-3 
50-3 
50-3 
60-3 
50-2 
50-2 
50-2 
50-2 
60.2 
50.2 

50.1 

60.1 
50.1 
50.1 
50.1 
50.0 
50.0 
50.0 
60.0 
50.0 
49.9 
49.9 
49.9 
49.9 
49.9 
49.9 
49.8 
49.8 
49.8 
49.8 
49.8 
49.7 
49.7 
49.7 
49.7 
49.7 
49.7 

10.274326 
274021 
273716 
273412 
273108 
272803 
272499 
272195 
271891 
271688 
271284 
10.270980 
270677 
270374 
270071 
269767 
269465 
269162 
268859 
268566 
268254 
10.267962 
267649 
267347 
267045 
266743 
266442 
266140 
265838 
265537 
265236 
10.264934 
264633 
264332 
264031 
263731 
263430 
263129 
262829: 
262529! 
262229 1 
10.261929 
261629. 
261329 
261029 1 
260729 j 
260430 
260130! 
259831! 
259532 1 
259233 
10.258934 
258635 
258336 
258038 
257739; 
257441 j 
257142 
256844 
256646 

266248! 

1 

! |4694' 
H46971 
! |46991 
j 4702- 
i 47051 
47071 
147101 
I 1 47121 
!47153 
4717S 
47204 
47229 
47255 
1 47281 
47306 
47332 
47358 
47383 
47409 
47434 
47460 
47486 
47511 
47537 
47562 
47588 
47614 
47639 
47665 
47690 
147716 

147741 
47767 

1 47793 
'47818 
47844 
47869 
47895 
47920 
47946 
47971 
47997 
48022 
48048 
48073 
48099 
48124 
48150 
48175 
48201 
48226 
48252 
482771 
48303 
48328! 
48354! 
48379! 
48405 
48430 
48456 
48481 

7 88295 
188281 
188267 
88254 
88240 
88226 
88213 
88199 
88185 
88172 
88158 
88144 
88130 
88117 
88103 
88089 
88075 
88062 
88048 
88034 
88020 
88006 
87993 
87979 
87965 
87951 
87937 
87923 
87909 
87896 
87882 
87868 
87854 
87840 
87826 
87812 
87798 
87784 
87770 
87756 
87743 
87729 
87715 
87701 
87687 
87673 
87659 
B7645 
37631 
37617 
37603 
37589 
37575 
37561 
37546 
37532 
37518 
37504 
■*7490 
'7476 
37462 


Cosine. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

f ~ 

61 Degrees. 

























































































50 Log. Sines and Tangents. (29°) Natural Sines. TABLE II. 


• 

1 

Sine. 

D. 10"| 

Cosine. 

D. 10" 

Tang. 

D. 10" 

Cotang. 

N. sine. 

N. cos. 


0 

1 

9.685571 

685799 

38.0 

37.9 

9.941819 

941749 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.7 

11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.8 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 
11.9 

11 Q 

9.743752 

744050 

49.6 
49.6 
49 6 
49.6 
49.6 
49.6 
49.5 
49.5 
49.5 

49.5 

49.6 
49.4 
49.4 
49.4 
49.4 
49.4 
49.3 
49.3 
49.3 
49.3 
49.3 
49.3 
49.2 
49.2 
49.2 
49.2 
49.2 
49.2 
49.1 
49.1 

AQ 1 

10.256248 

255960 

48481 

48506 

87462 

87448 

60 

69 

2 

686027 

941679 

744348 

255652 

48532 

87434 

58 

3 

4 

686254 

686482 

37.9 

37.9 

941609 

941539 

744646 

744943 

255355 

255067 

48557 

48583 

87420 

87406 

57 

56 

5 

686709 

37.9 

37.8 

37.8 

37.8 

37.8 
37.7 
37.7 
37.7 

37.7 
37.6 
37.6 
37.6 
37.6 

37.6 
37.5 
37.5 

37.5 
37.4 
37.4 
37.4 

37.4 
37.3 
37.3 
37.3 
37.3 
37.2 
37.2 
37.2 
37.1 
37.1 
37.1 
37.1 
37.0 
37.0 
37.0 
37.0 

36.9 
36.9 
36.9 
36.9 

36.8 
36.8 
36.8 
36.8 

36.7 
36.7 
36.7 
36.7 

36.6 
36.6 
36.6 
36.6 

36.5 

36.6 
36.5 
36.5 

941469 

745240 

254760 

48608 

87391 

65 

6 

686936 

941398 

746538 

254462 

48634 

87377 

64 

7 

687163 

941328 

745835 

254166 

48659 

87363 

63 

8 

687389 

941258 

746132 

253868 

48684 

87349 

52 

9 

687616 

941187 

746429 

253571 

48710 

87335 

51 

10 

687843 

941117 

746726 

253274 

48736 

87321 

50 

11 

9 .688069 

9.941046 

9.747023 

10.252977 

48761 

87306 

49 

12 

688295 

940975 

747319 

252681 

48786 

87292 

48 

13 

688521 

940905 

747616 

252384 

48811 

87278 

47 

14 

688747 

940834 

747913 

252087 

48837 

87264 

46 

15 

688972 

940763 

748209 

251791 

48862 

87250 

45 

16 

689198 

940693 

748506 

251495 

48888 

87235 

44 

17 

689423 

940622 

748801 

251199 

48913 

87221 

43 

18 

689648 

940551 

749097 

250903 

48938 

87207 

42 

19 

689873 

940480 

749393 

250607 

48964 

87193 

41 

20 

690098 

940409 

749689 

260311 

48989 

87178 

40 

21 

9.690323 

9.940338 

9.749985 

10.250015 

49014 

87164 

39 

22 

690548 

940267 

750281 

249719 

49040 

87150 

38 

23 

690772 

940196 

750576 

249424 

49065 

87136 

37 

24 

690996 

940125 

750872 

249128 

49090 

87121 

36 

25 

691220 

940054 

751167 

248833 

49116 

87107 

35 

26 

691444 

939982 

751462 

248538 

49141 

87093 

34 

27 

691668 

939911 

751767 

248243 

49166 

87079 

33 

28 

691892 

939840 

752052 

247948 

49192 

87064 

32 

29 

692115 

939768 

752347 

247653 

49217 

87050 

31 

30 

692339 

939697 

762642 

247358 

49242 

87036 

30 

31 

9.692562 

9.939625 

9.752937 

49 1 

10.247063 

49268 

87021 

29 

32 

692785 

939554 

763231 

49.1 

49.1 
49.0 
49.0 
49.0 
49.0 
49.0 
49.0 
48.9 
48.9 
48 9 

246769 

49293 

87007 

28 

33 

693008 

939482 

753526 

246474 

49318 

86993 

27 

34 

693231 

939410 

753820 

246180 

49344(86978 

26 

35 

693453 

939339 

754115 

245885 

49369)86964 

25 

36 

693676 

939267 

12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.0 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12.1 
12 1 

754409 

245591 

4939486949 

24 

37 

693898 

939195 

754703 

245297 

t 49419 86935 

23 

38 

694120 

939123 

754997 

245003 

149445 86921 

22 

39 

694342 

939052 

765291 

244709 

49470 86906 

21 

40 

694564 

938980 

755585 

244415! 

149495186892 

20 

41 

9.694786 

9.938908 

9.755878 

10.244122 

14952186878 

19 

42 

695007 

938836 

756172 

243828 

! 49546 86863 

18 

43 

695229 

938763 

756466 

48.9 
48 9 

243636 

49571 86849 

17 

44 

695450 

938691 

756759 

243241 

' 49596186834 

16 

45 

695671 

938619 

767052 

48 9 

342948 

49622 

86820 

15 

46 

695892 

938547 

767345 

48 8 

242665 

149647 

86805 

14 

47 

696113 

938476 

757638 

48.8 
48.8 
48 8 

242362 

149672 

86791 

13 

48 

696334 

938402 

757931 

242069 

1 49697186777 

12 

49 

696554 

938330 

758224 

241776 

1 49723(86762 

11 

50 

696775 

938268 

768517 

48.8 
48.8 
48 7 

241483 

49748 86748 

10 

51 

9.696995 

9.938185 

9.758810 

10.241190 

49773 86733 

9 

52 

697215 

938113 

759102 

240898 

49798(86719 

8 

53 

697435 

938040 

769395 

48 7 

240605 

49824 86704 

7 

54 

697654 

937967 

759687 

48.7 
48 7 

240313 

49849 86690 

6 

55 

697874 

937895 

759979 

240021 

4987486675 

5 

56 

698094 

937822 

12.1 

12.1 

12.1 

12.1 

760272 

48.7 

48.7 

48.6 

48.6 

239728 

j 49899)86661 

4 

67 

698313 

937749 

760564 

239436 

1 4992486646 

3 

58 

698632 

937676 

760856 

239144 

49950 86632 

2 

59 

698751 

937604 

761148 

238852 

4997 £ 

.86617 

1 

60 

698970 

937531 

761439 

238661 

60001 

86603 

0 


Cosine. 


Sine. 


Co tang. 


Tang. 

N. cos 

. IN.sine 

/ 


60 Degrees. 





























































- TABLE II. 

Log. Sines and Tangents. (30°) Natural Sines 


51 

| 

feme. 

D. 10' 

' Cosine. 

D. 10' 

j Tan» 

D. 10' 

Cotaug. 

N. sine.IN. cos 


1 ° 

1 2 
1 3 

1 4 

5 

6 
7 

1 8 
1 s 

1 10 

1 11 
12 

13 

14 

15 
; i6 

n 

18 

19 

20 

21 

22 

23 

24 
26 
26 

1 27 

1 23 

I 29 
30 

1 31 

I 32 

33 

1 34 

35 

36 

37 
| 33 

39 

I 49 

I 41 

I 42 

1 43 

1 44 

1 45 

46 

1 47 

I 48 

! 49 

I I 50 

1 51 r 

I 62 

53 
641 

1 ^ 

1 

57 

58 

59 
60 

9.693970 

699189 

6994)7 

699626 

699844 

70JJ62 

700280 

700498 

700716 

700933 

701151 

9.701368 

701585 

701802 

702019 

702236 

702452 

702669 

702885 

703101 

703317 

9.703533 

703749 

703964 

704179 

704395 

704610 

704825 

705040 

705254 

705469 

9.705683 

705898 

706112 

706326 

708539 

706753 

706967 

707180 

707393 

707606 

9.707819 

708032 

708245 

708458 

7046/0 

708882 

700004 

700308 

700518 

709730 

1.709941 

710153 

710j64 

710575 

710786 

710967 

711208 

711419 

711629 

711839 

(36.4 

136.4 
36.4 
36.4 

136.3 

135.3 
36.3 

i 36.3 

36.3 
36.2 
36.2 
36.2 
36.2 
36.1 
36.1 

i 36.1 
j 36.1 
36.0 
36.0 
36.0 
36.0 
35.9 
35.9 
35.9 
35.9 
35.9 
35.8 
35.8 
35.8 
35.8 
35.7 
35.7 
35.7 
35.7 
35.6 
35.6 
35.6 
35.6 
35.5 
35.5 
35.5 
35.5 

35.4 
35.4 
35.4 
35.4 
35.3 
35.3 
35.3 
35.3 
35.3 
35.2 
35.2 
35.2 
35.2 
35.1 
35.1 
35.1 
35.1 
35.0 

9.937531 
937458 
937385 
937312 
937238 
937165 
937092 
937019 
936946 
936872 
936799 
9.936725 
936652 
936578 
936505 
936431 
936357 
936284 
936210 
936136 
936062 
9.935988 
935914 
935840 
935766 
935692 
935618 
935543 
935469 
935395 
935320 
9.935245 
935171 
935097 
935022 
934948 
934873 
934798 
934723 
934649 
934574 
9.934499 
934424 
934349 
934274 
934199 
934123 
934048 
933973 
933898 
933822 
9.933747 
933671 
933596 
933520 
933445 
933369 
933293 
933217 
933141 
933066 

12.1 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.2 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.3 

12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 
12.4 

12.4 

12.5 
12.5 
12.5 
12.5 
12.5 

12.5 

12.6 
12.6 
12.5 
12.5 
12.5 

12.5 

12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 
12.6 

9.761439 

761731 

762023 

762314 

762605 

76289/ 

763188 

763479 

763770 

764051 

764352 

9.764643 

764933 

765224 

765514 

765805 

766095 

766385 

766675 

766965 

767255 

9.767545 

767834 

768124 

768413 

768703 

768992 

769281 

769570 

769860 

770148 

9.770137 

770726 

771015 

771303 

771592 

771880 

772168 

772457 

772745 

773033 

9.773321 

773608 

773896 

774184 

774471 

774759 

776046 

775333 

775621 

775908 

9.776195 

776482 

776769 

777055 

777342 

777628 

777915 

778201 

778487 

778774 

48.4 
48.6 
48.6 
48.6 

48.5 
48.5 
48.5 
48.5 
48.5 
48.5 
48.4 
48.4 
48.4 
48.4 
48.4 
48.4 
48.4 
48.3 
48.3 
48.3 
48.3 
48.3 
48.3 
48.2 
48.2 
48.2 
48.2 
48.2 
48.2 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.1 
48.0 
48.0 
48.0 
48.0 
48.0 
48.0 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.9 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47.8 
47.7 
47.7 
47.7 
47.7 

10.238561 
238269 
23/977 
237686 
237394 
237103 
236812 
236521 
236230 
235939 
235648 
10.235357 
235067 
234776 
234486 
234195 
233905 
233616 
233325 
233035 
232745 
10.232455 
232166 
231876 
231587 
231297 
231008 
230719 
230430 
230140 
229852 
10-229563 
229274 
228985 
228697 
228408 
228120 
227832 
227543 
227255 
226967 
10-226679 
226392 
226104 
225816 
225529 
225241 
224954 
224667 
224379 
224092 | 
10-223805i 
223518! 
223231 
222945 
222658 
222372 
222085 
221799 
221612 
221226 

50001 
50027 
5005, 
50071 
50101 
5012o 
50151 
60176 
50201 
50227 
i 50252 
50277 
50302 
5032/ 
50352 
50377 
50403 
60428 
50453 
50478 
60503 
50528 
j 50553 
!50678 
50603 
50628 
60654 
50679 
60704 
50729 
50754 
50779 
50804 
50829 
50854 
50879 
60904 
50929 
50954 
50979 
51004 
51029 
51054 
51079 
51104 
61129 
51154 
51179 
51204 
51229 
51254) 
61279; 
51304) 
51329) 
51354) 
51379) 
61404) 
51429) 
51454) 
51479) 
51504) 

36603 
86588 
865/3 
86559 
86544 
86530 
86515 
86501 
86486 
86471 
86457 
86442 
86427 
86413 
86398 
86384 
86369 
86354 
86340 
86325 
86310 
86295 
86281 
86266 
86251 
86237 
86222 
86207 
86192 
86178 
86163 
86148 
86133 
86119 
861(K 
86089 
860/4 
86059 
86045 
86030 
86015 
86000 
85985 
85970 
85956 
85941 
85926 
65911 
65896 
65881 
65866 
65851 
65836 
65821 
65806 
65792 
66777 
65762 
65747 
65732 
65717 

60 

59 

58 

57 

56 

55 

54 

63 

52 

61 

60 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 

“1 

Cosine. 


Sine 


Co tang. 


Tang. 

N. cos. 

N.sine. 

t 

59 Degrees. 






















































































52 


Log. Sines and Tangents. (31°) Xataral Sines. TABLE II. 


N.sine. N. cos. 


1 

Sine. 

D. 10" 

Cosine. 

0 

9.711839 

35.0 

35.0 

35.0 

34.9 

34.9 

34.9 

34.9 

34.9 

34.8 

34.8 

34.8 

34.8 
34.7 
34.7 
34.7 
34.7 

34.7 
34.6 
34.6 
34.6 
34.6 
34.6 
34.6 

34.6 
34.5 
34.5 
34.4 
34.4 
34.4 
34.4 
34.3 
34.3 
34.3 
34.3 
34.3 
34.2 
34.2 
34.2 
34.2 
34.1 
34.1 
34.1 
34.1 
34.0 
34.0 
34 0 
34.0 
34.0 

33.9 
33.9 
33.9 
33.9 
33.9 

33.8 
33-8 
33.8 
33.8 

33.7 
33.7 
33.7 

9.933036 

1 

712050 

932990 

2 

712260 

932914 

3 

712469 

932838 

4 

712679 

932/62 

6 

712889 

932685 

6 

713098 

932609 

7 

713308 

932533 

8 

713517 

932457 

9 

713726 

932380 

1 10 

713935 

932304 

1 11 

9.714144 

9.932228 

t 12 

714352 

932151 

13 

714561 

932075 

14 

714769 

931998 

15 

714978 

931921 

16 

715186 

931845 

17 

715394 

931768 

18 

715602 

931691 

19 

715809 

931614 

20 

710017 

931537 

21 

9.716224 

9.931460 

22 

716432 

931383 

23 

716039 

931306 

24 

716846 

931229 

25 

717053 

931152 

26 

717269 

931075 

27 

717466 

930998 

28 

717673 

930921 

29 

717879 

930843 

30 

718085 

930766 

31 

9.718291 

9.930688 

32 

718497 

930611 

33 

718703 

930533 

34 

718909 

930456 

36 

719114 

930378 

36 

719320 

930300 

37 

719525 

930223 

38 

719730 

930145 

39 

719936 

930067 

40 

720140 

929989 

41 

9.720345 

9.929911 

42 

720549 

929833 

43 

720754 

929755 

44 

720958 

929677 

45 

721162 

929599 

46 

721366 

929521 

47 

721570 

929442 

48 

721774 

929364 

49 

721978 

929286 

50 

722181 

929207 

51 

9.722385 

9.929129 

62 

722588 

929060 

63 

64 

722791 

722994 

928972 

928893 

55 

723197 

928815 

66 

723400 

928736 

57 

723603 

928667 

68 

723805 

928578 

69 

724007 

928499 

60 

724210 

928420 


Cosine. 

Sine. 


D. 10" 


12.6 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.7 

12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.8 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
12.9 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.0 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 
13.1 


Tang. 


1.778774 

779060 

779346 

779632 

779918 

780203 

780489 

780776 

781060 

781346 

781631 

1.781916 

782201 

782486 

782771 

783066 

783341 

783626 

783910 

784195 

784479 

.784764 

785048 

785332 

785616 

786900 

786184 

786468 

786762 

787036 

787319 

.787603 

787886 

788170 

788453 

788736 

789019 

789302 

789586 

789868 

790151 

.790433 

790716 

790999 

791281 

791663 

791846 

792128 

792410 

792692 

792974 

.793266 

793538 

793819 

794101 

794383 

794664 

794946 

796227 

795608 

795789 


Cotang. 


D. 10" 


47.7 

47.7 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 
47.6 

47.5 

47.6 
47.6 
47.5 
47.4 
47.4 
47.4 
47.4 
47 
47 
47 
47 
47.3 
47.3 
47.3 
47.3 
47.3 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.2 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.1 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
47.0 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 
46.9 

46.8 


Co tang. 


10.221226 
220940 
220654 
220368 
220082 
219797 
219511 
219225 
218940 
218654 
218369 
10.218084 
217799 
217614 
217229 
216944 
216659 
216374 
216090 
215805 
216521 
10.216236 
214962 
214668 
214384 
214100 
213816 
213632 
213248 
212964 
212681 
10.212397 
212114 
211830 
211547 
211264 
210981 
210698 
210416 
210132 
209849 
10.209567 
209284 
209001 
208719 
208437 
208164 
207872 
207690 
207308 
207026 
10.206744 
206462 
206181 
205899 
205617 
205336 
205055 
204773 
204492 
204211 


61504 86717 60 
61629 86702 
51554 85687 
51579 85672 
51004 85657 
61628 85642 
61663 85627 
51678 85612 
51703 85597 


85582 _ 
85567 60 
85551 49 

I _ _ .. 


Tang. 


N. cos 


69 

58 

67 

56 

55 

64 

63 

62 

51 


51728 
61753 

61778_ 

61803185536 48 


61828 85521 47 
61852 85506 46 
61877 85491 45 
61902 85476 44 
61927 85461 43 
61962 85440 42 
i ; 61977 85431 41 
! '62002 85416 40 
62026185401 39 
6205185385 38 
62076 85370 37 
62101 [85365 36 
62126 85340 36 
62161 85325 34 
52175 85310 33 
62200 85294 32 
62226 86279 31 
62250 86204 3( 
62276i85249 21 
62299 85234 2f 
62324 85218 O' 
62349 85203 
62374 85188 
62399 85173 
52423 85157 
62448 85142 
62473 85127 
62498 85112 
62522 86090 
62547 85081 
62672 85066 
6259 7 85061 
62021 86036 
62040 86020 
62671 85006 
62690 84989 
62720 84974 
62746 84959 
62770 84943 
152794 84928 
J 52819 84913 
62844 84897 
52809 84882 
62893 84806 
62918 8486 * 
629431'"" 

62967J 
629921 


N.siue. 


58 Degrees. 






























































TABLE Tl. 


Log, Sines and Tangents. 


T 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

3 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 
51 
62 
63 
54 

65 

66 
67 

58 

59 

60 


Sine. 


9.724210 
724412 
724614 
724816 
725017 
725219 
725420 
725622 
725823 
726024 
726225 
9.726426 
726626 
726827 
727027 
727228 
727428 
727628 
727828 
728027 
728227 
9.728427 
728626 
728825 
729024 
729223 
729422 
729621 
729820 
730018 
730216 
9.730415 
730613 
730811 
731009 
731206 
731404 
731602 
731799 
731996 
732193 
9 732390 
732587 
732784 
732980 
733177 
733373 
733569 
733765 
733961 
734157 
9 734353 
734549 
734744 
734939 
735135 
735330 
735525 
735719 
735914 
736109 


D. 10' 


33.7 

33.7 
33.6 
33.6 
33.6 
33.6 

33.5 

33.6 
33.5 
33.5 

33.5 
33.4 
33.4 
33.4 
33.4 

33.4 
33.3 
33.3 
33.3 
33.3 
33.3 
33.2 
33.2 
33.2 
33.2 
33.1 
33.1 
33.1 
33.1 
33.0 
33.0 
33.0 
33.0 
33.0 
32.9 
32.9 
32.9 
32.9 
32.9 

32.8 
32.8 
32.8 
32.8 
32.8 

32.7 
32.7 
32.7 
32.7 
32.7 

32.6 
32.6 
32.6 
32.6 

32.5 

32.5 

32.6 

32.5 

32.6 
32.4 
32.4 


Cosine. |D. 10' 


9.928420 
928342 
928263 
928183 
928104 
928025 
927946 
927867 
927787 
927708 
927629 
9.927549 
927470 
927390 
927310 
927231 
927151 
927071 
926991 
926911 
926831 
9.926751 
926671 
926591 
926511 
926431 
926351 
926270 
926190 
926110 
926029 
9.925949 
925868 
925788 
925707 
925626 
925545 
926465 
925384 
925303 
925222 
9.925141 
926060 
924979 
924897 
924816 
924735 
924654 
924572 
924491 
924409 
9.924328 
924246 
924164 
924083 
924001 
923919 
923837 
923755 
923673 
923591 


Cosine. 


Sine. 


13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.2 

13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
13.3 
J3.3 

13.3 

13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 
13.4 

13.4 

13.5 

13.5 

13.6 
13.5 
13.5 
13.5 
13.5 

13.5 

13.6 

13.5 

13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 
13.6 

13.6 

13.7 
13.7 


Tang. 


(32°) Natural Sines. 
' D. 10" 


53 


9.795789 
796070 
796351 
798632 
796913 
797194 
797475 
797755 
798036 
798316 
798596 
9.^98877 
799167 
799437 
799717 
799997 
800277 
800557 
800836 
801116 
801396 
9.801675 
801955 
802234 
802513 
802792 
803072 
803351 
803630 
803908 
804187 
9.804466 
804745 
805023 
805302 
805580 
806859 
806137 
806415 
806693 
806971 
9.807249 
807627 
807805 
808083 
808361 
808638 
808916 
809193 
809471 
809748 
810025 
810302i 
810580 
810857 
811134 
811410 
811687 
811964 
812241 
812517 


Cotang. 


46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46.8 

46.7 

46.7 

46.7 

46.7 

46.7 

46.7 

46.7 

46.6 

46.6 

46.6 

46.6 

46.6 

46.6 

46.6 

46.6 

46.5 

46.5 

46.5 

46.5 

46.5 

46.5 

46.5 

46.5 

46.4 

46.4 

46.4 

46.4 

46.4 

46.4 

46.4 

46.3 

46.3 

46.3 

46.3 

46.3 

46.3 

46.3 

46.3 

46.2 

46.2 

46.2 

46.2 

46.2 

46.2 

46.2 

46.2 

46.2 

46.1 

46.1 

46.1 

46.1 

46.1 


j Cotang. 

N. sine. 

N. cos. 


10.204211 

‘52992 

84805 

60 

203930 

63017 

84789 

69 

203649 

53041 

84774 

68 

203368 

63086 

84759 

57 

203087 

53091 

84743 

56 

202806 

53115 

84728 

55 

212625 

53140 

84712 

54 

202245 

63164 

84697 

63 

201964 

53189 

84681 

52 

201684 

53214 

84666 

61 

201404 

53238 

84650 

50 

10.201123 

53263 

84635 

49 

200843 

63288 

84619 

48 

200563 

53312 

84604 

47 

200283 

63337 

84588 

46 

200003 

53361 

84573 

46 

199723 

53386 

84567 

44 

199443 

53411 

84542 

43 

199164 

i53435 

84526 

42 

198884 

63460 

84511 

41 

198604 

63484 

84495 

40 

10.198325 

53509 

84480 

39 

198046 

53534 

84464 

38 

197766 

53658 

84448 

37 

197487 

63583 

84433 

36 

197208 

63607 

84417 

35 

196928 

63632 

84402 

34 

196649 

63656 

84386 

33 

196370 

63681 

84370 

32 

196092 

63705 

84355 

31 

195813 

53730 

84339 

30 

10.195534 

63754 

84324 

29 

195255 

53779 

84308 

28 

194977 

63804 

84292 

27 

194698 

53b28 84277 

26 

194420 

63853]84261 

25 

194141 

53877 

84245 

24 

193863 

53902 84230 

23 

193586 

5392684214 

22 

193307 

53951,84198 

21 

193029 

5397584182 

20 

10.192751 

6400084167 

19 

192473 

54024 84161 

18 

192195 

54049 84135 

17 

191917 

6407384120 

16 

191639 

54097|84104 

15 

191362 

54122 84088 

14 

491084i 

64146184072 

13 

190807 

54171 

84057 

12 

190529 

64195 

84041 

11 

190262 

64220 

84025 

10 

10.189975 

54244 

84009 

9 

189698 

64269 

83994 

8 

189420 

64293 

83978 

7 

189143 

54317 

83962 

6 

188866 

5434283946 

5 

188590 

5436683930 

4 

188313 

64691183916 

3 

188036 

64415183899 

2 

187759 

64440 

83883 

1 

187483 

64464 

83867 

0 

Tang. 

N. cos. 

N.sine. 

r 


57 Degrees. 




































































I 

54 Log. Sines and Tangents. (33°) Natural Sines. TABLE II. 


/ 

Sine. 

D. 10' 

Cosine. 

|D. 10' 

Tang. 

D. 10' 

Cotang. 

N. sine 

. N. cos 


0 

9.736109 

32.4 

32.4 

32.4 

32.3 

32.3 

32.3 

32.3 

9.923591 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

13.7 

9.812517 

46.1 
46.1 
46 1 
46.0 
46.0 
46.0 
46.0 

10.187482 

!54464 

83867 

60 

1 

736303 

923509 

812794 

• 187206 

1164488 

83851 

59 

2 

736498 

923427 

813070 

’86930 

64513 

83836 

58 

3 

736692 

923345 

813347 

186653 

j 54537 

83819 

57 

4 

736886 

923263 

813623 

186377 

164561 

83804 

56 

6 

737080 

923181 

813899 

186101 

64586 

83788 

55 

6 

737274 

923098 

814175 

185826 

64610 

83772 

54 

7 

737467 

923016 

814452 

185548 

64635 

83766 

53 

8 

737661 

32.3 
32.2 
32.2 
32.2 
32.2 
32.2 
32.1 
32.1 
32.1 
32.1 
32.1 
32.0 
32.0 
32.0 
32.0 
32.0 
31.9 
31.9 
31.9 
31.9 
31.9 
31.8 

922933 

13.7 

13.7 

13.7 

13.8 
13.8 
13.8 
13.8 
13.8 
13.8 
13.8 
13.8 
13.8 
13.8 

13.8 

13.9 
13.9 
13.9 
13.9 
13.9 
13.9 
13.9 
13.9 

814728 

46.0 
46.0 
46.0 
46 0 

185272 

64659183740 

62 j 

9 

737855 

922851 

815004 

184996 

64683183724 

51 

10 

738048 

922768 

816279 

184721 

54708 

83708 

50 

11 

9.738241 

9.922686 

9.815555 

10.184445 

64732 

83692 

49 

12 

738434 

922603 

815831 

41 . y 

45.9 

46.9 
46 9 

45.9 
45.9 
45.9 
45.9 
45.9 
45.8 

45.8 

46.8 

46.8 

45.8 
45.8 
45.8 
45.8 
45.8 

184169 

64756 

83676 

48 

13 

738627 

922620 

816107 

183893 

64781183660 

47 

14 

738820 

922438 

816382 

183618 

54805 

83645 

46 

15 

739013 

922355 

816658 

183342 

64829 

83629 

45 

16 

739206 

922272 

816933 

183067 

64864 

83613 

44 

17 

739398 

922189 

817209 

182791 

6487883597 

43 

18 

739590 

922106 

817484 

182516 

i 6490283681 

42 

19 

739783 

922023 

817759 

182241 

64927 83565 

41 

20 

739975 

921940 

818036 

181966 

154951 83549 

40 

21 

9.740167 

9.921857 

9.818310 

10.181690 

64975 83533 

39 

22 

740359 

921774 

818585 

181416 

164999 83517 

38 

23 

740550 

921691 

818860 

181140 

| 65024 83501 

37 

24 

740/42 

921607 

819135 

180865 

165048 83485 

36 

25 

740934 

921524 

819410 

180690 

55072|83469 

35 

26 

741125 

921441 

819684 

180316 

65097 83453 

34 

27 

741316 

921357 

819959 

180041 

55121 ;83437 

33 

28 

741508 

921274 

820234 

179766 

6514583421 

32 

29 

741699 

921190 

820508 

179492 

55169 83405 

31 

30 

741889 

31.8 
31.8 
31.8 
31.8 
31.7 
31.7 
31.7 
31.7 
31.7 
31.6 
31.6 
31.6 
31.6 
31.6 
31.5 
31.5 
31.5 
31.5 
31.5 
31.4 
31.4 
31.4 
31.4 
31.4 
31.3 
31 3 

921107 

13.9 

13.9 

13.9 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.0 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.1 

14.2 
14.2 
14.2 
14.2 

820783 

45.7 
45.7 
45.7 
45.7 

45.7 

46.7 

46.7 

45.7 
45.7 
45.6 
45.6 

45.6 

46.6 

45.6 
45.6 
45.6 
45.6 
45.6 
45.6 

45.5 

46.5 

45.6 

45.5 

45.6 

45.5 

45.6 
45.5 

46.4 

45.4 

46.4 

45.4 

179217 

55194 83389 

30 

31 

9.742080 

9.921023 

9.821057 

10.178943 

5521883373 

29 

32 

742271 

920939 

821332 

178668 

55*42 83356 

28 

33 

742462 

920856 

821606 

178394 

5526683340 

27 

34 

742652 

920772 

821880 

178120 

55291 83324 

26 

35 

742842 

920688 

822164 

177846 

65315 83308 

25 

36 

743033 

920604 

822429 

177671 

65339 

83292 

24 

37 

743223 

920520 

822703 

177297 

65363 

83276 

23 

38 

743413 

920436 

822977 

177023 

65388 

83260 

22 

39 

743602 

920352 

823250 

176750 

55412 

63244 

21 

40 

743792 

920268 

823524 

176476 

55436 

83228 

20 

41 

9.743982 

9.920184 

9.823798 

10.176202 

55460 

63212 

19 

42 

744171 

920099 

824072 

175928 

55484 83196 

18 

43 

744361 

920015 

824345 

176655 

5550983179 

17 

44 

744550 

919931 

824619 

175381 

555o3 j83163 

16 

45 

744739 

919846 

824893 

176107 

55557 83147 

15 

46 

744928 

919762 

825166 

174834 

5568183131 

14 

47 

745117 

919677 

825439 

174561 

55605 83115 

13 

48 

745306 

919593 

826713 

1742871 

55630 83098 

12 

49 

50 

51 

745494 

745683 

9.745871 

919508 

919424 

9.919339 

825986 

826259 

9.826532 

174014, 
173741 | 
10.173468 

5565 1 83082 
55b78 830o6 
5570283050 

11 

10 

9 

52 

746059 

919264 

826805 

173196 

55726183034 

8 

53 

746248 

919169 

827078 

172922 

55760 83017 

7 

54 

746436 

919085 

827351 

172649 

65776 83001 

6 

55 

746624 

31.3 
31.3 
31 3 

919000 

827624 

172376 

55799 

62585 

5 

56 

746812 

918915 

827897 

172103 

55823 

62569 

4 

57 

746999 

918830 

828170 

171830 

65847!: 

62963 

3 

58 

747187 

31 2 

918745 

828442 

171668 

65871 

52936 

2 

59 

747374 

31.2 

9J8b59 

828716 

171285 

55695 82920 

1 

60 

747562 

918574 

828987 

171013 

55919? 

62904 

0 


Cosii i. 


Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

/ 


56 Degrees. 



































































TABLE II. Log. Sines and Tangents. (34°) Natural Sines. 


55 


0 
1 
2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 
33 

54 

55 

56 

57 

58 
'.9 
60 


Sine. 

9.747562 
747749 
747936 
748123 
748310 
748497 
74S683 
748870 
749036 
749243 
749426 
9.749615 
749801 
749987 
750172 
750358 
750543 
750729 
750914 
751099 
751284 
.751469 
751654 
751839 
752023 
762208 
752392 
762576 
752760 
752944 
763128 
763312 
753495 
753679 
753862 
754046 
754229 


TOO", Cosiue. 


31.2 

31.2 

31.2 


31.1 
31.1 
31.1 
31.1 
31.1 
31.0 
31.0 
31.0 
31.0 
31.0 
30.9 
30.9 
30.9 
30.9 
30.9 
30.8 
30.8 
30.8 
30.8 
30.8 
30.8 
30.7 
30.7 
30.7 
30.7 
; 30.7 
30.6 
30.6 
,30.6 
,30.6 
|30.6 
30.6 
30.5 

75441° 

306 


I K 


754778 
754960 
9.765143 
755326 
755508 
755690 
755872 
766054 
756236 
756418 
756600 
766782 
766963 
757144 
757326 
757507 
757688 
757869 
758050 
758230 
758411 
768591 


30.5 

30.4 

30.4 

30.4 

30.4 

30.4 

30.4 

30.3 

30.3 

30.3 

30.3 

30.3 

30.2 

30.2 

30.2 

30.2 

30.2 

30.1 

30.1 

30.1 

30.1 

30.1 


'9.918574 
918489 
918404 
918318 
918233 
918147 
918052 
917976 
917891 
917805 
917719 
9.917634 
917548 
917462 
917376 
917290 
917204 
917118 
917032 
916946 
916859 
9.916773 
916687 
916600 
916514 
916427 
916341 
916254 
916167 
916081 
915994 
9.915907 
915820 
915733 
915646 
915659 
915472 
915385 
915297 
915210 
915123 
915036 
914948 
914860 
914773 
914685 
914598 
914510 
914422 
914334 
914246 
9.914168 
914070 
913982 
913894 
913806 
913718 
913630 
913541 
913453 
913365 


D. 10" 


Cosine. 1 


Sine. 


14.2 
14.2 
14.2 
14 
14.2 

14.2 
14 
14 
14 
14 

14.3 
14.3 
14.3 
14.3 
14.3 
14.3 

14.3 

14.4 
14.4 
14.4 
14.4 
14.4 
14.4 
14.4 
14.4 
14.4 
14.4 

14.4 

14.5 

14.5 

14.6 

14.5 

14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 
14.6 

14.6 

14.7 
14.7 
14.7 
14.7 
14.7 
14.7 
14.7 
14.7 
14.7 
14.7 


Tang. 


9. 


9.828987 
829260 
829532 
829805 
830077 
830349 
830521 
830893 
831165 
831437 
831709 
831981 
832253 
832525 
832796 
833068 
833339 
833611 
833882 
834154 
834425 
9.834696 
834967 
835238 
835509 
835780 
836051 
836322 
836593 
836864 
837134 
9.837405 
837676 
837946 
838216 
838487 
838767 
839027 
839297 
839568 
839838 
9.840108 
840378 
840647 
840917 
841187 
841467 
841726 
841996 
842266 
842536 
9.842805 
843074 
843343 
843612 
843882 
844151 
844420 
844689 
844958 
845227 


D. 10" 


45.4 

45.4 

45.4 

45.4 

45.4 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.3 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

45.2 

46.2 

45.2 
45.2 
45.1 
45.1 
45.1 
45.1 
45.1 
45.1 
45.1 
45.1 
45.1 
45.1 


Cotang. I ;N .sine 


10.171013 li 55919 


10 . 


45 
45 
45 
45 
45 
45 
45.0 
45.0 
45.0 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.9 
44.8 
44.8 
44.8 
44.8 
44.8 


Cotang. 


170740 
170468 
170195 
169923 
169651 
169379 
169107 
168836 
168563 
168291 
168019 
167747 
167475 
167204 
166932 
166661 
166389 
166118 
165846 
165575 
10.165304 
165033 
164762 
164491 
164220 
163949 
163678 
163407 
163136 
162866 
10.162595 
162326 
162054 
161784 
161513 
161243 
160973 
160703 
160432 
160162 
10.159892 
169622 
159353 
169083 
158813 
158543 j 
158274i 
158004 
157734 
167466 
157195 
156926 
156657 
156388 
166118 
155849 
155580 
155311 
165042 
154773 


10 


Tang. 


i55943 
j55968 
i56992 
56016 
l56040 
56064 
66088 
56112 
66136 
56160 
56184 
56208 
56232 
56256 
66280 
56305 
66329 
66363 
66377 
66401 
56425 
66449 
66473 
56497 
66521 
56545 
56569 
66593 
56617 
66641 
66665 
56689 
66713 
56736 
56760 
66784 
66808 
66832 
66856 
66880 
56904 
56928 
56952 
66976 
57000 
67024 
57047 
67071 
67095 
57119 
57143 
57167 
57191 
57216 
67238 
57262 
57286 
57310 
57334 
57358 


N. COS.; 


82904 

82887 

82871 

82855 

82839 

82822 

82806 

82790 

82773 

82757 

82741 

82724 

82708 

82692 

82675 

82659 

82643 

82626 

82610 

82593 

82577 

82561 

82544 

82528 

82511 

82495 

82478 

82462 

82446 

82429 

82413 

82396 

82380 

82363 

82347 

82330 

82314 

82297 

82281 

82264 

82248 

82231 

82214 

82198 

82181 

82166 

82148 

82132 

82115 

82098 

82082 

82066 

82048 

82032 

82015 

81999 

81982 

81965 

81949 

81932 

81915 


N. cos. N.sine. 


60 

59 

68 

57 

56 

55 

54 

63 

52 

51 

50 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


55 Degrees. 


















































































1 


56 Log. Sines and Tangents. (35°) Natural Sines. TABLE II. 


9 

Sine. 

D. 10' 

Cosine. 

L). 10' 

Tang. 

D. 10" 

Cotang. N. sine. 

N. cos. 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 
46 

46 

47 

48 

49 

50 
61 

52 

53 

54 

65 

66 
67 

58 

59 

60 

9.758591 

758772 

758952 

759132 

759312 

759492 

759672 

769852 

760031 

760211 

760390 

9.760569 

760748 

760927 

761106 

761285 

761464 

761642 

761821 

761999 

762177 

9.762356 

762534 

762712 

762889 

763067 

763246 

763422 

763600 

763777 

763954 

9.764131 

764308 

764485 

764662 

764838 

765015 

765191 

765367 

765544 

765720 

9.765896 

766072 

766247 

766423 

766598 

766774 

766949 

767124 

767300 

767475 

9.767649 

767824 

767999 

768173 

768348 

768522 

768697 

768871 

769045 

769219 

30.1 
30.0 
30.0 
30.0 
30.0 
30.0 
29.9 
29.9 
29.9 
29.9 
29.9 
29.8 
29.8 
29.8 
29.8 
29.8 
29.8 
29.7 
29.7 
29.7 
29.7 
29.7 
29.6 
29.6 
29.6 
29.6 
29.6 
29.6 
29.5 
29.5 
29.5 
29.5 
29.5 
29.4 
29.4 
29.4 
29.4 
29.4 
29.4 
29.3 
29.3 
29.3 
29.3 
29.3 
29.3 

29.2 
29.2 
29.2 
29.2 
29.2 
29.1 
29.1 
29.1 
29.1 
29.1 
29.0 
29.0 
29.0 
29.0 
29.0 

3.913365 
913276 
913187 
913099 
913010 
912922 
912833 
912744 
912655 
912566 
912477 
9.912388 
912299 
912210 
912121 
912031 
911942 
911853 
911763 
911674 
911584 
9.911495 
911405 
911315 
911226 
911136 
9H046 
910956 
910866 
910776 
910686 
9.910596 
910506 
910415 
910325 
910236 
910144 
910054 
909963 
909873 
909782 
9.909691 
909601 
909510 
909419 
909328 
909237 
909146 
909055 
908964 
908873 
9.908781 
908690 
9085 9 
908507 
908416 
908324 
908233 
908141 
908049 
907958 

14.7 

14.7 

14.8 
14.8 
14.8 
14.8 
14.8 
14.8 
14.8 
14.8 
14.8 

14.8 

14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
14.9 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.0 
15.1 
15.1 
15.1 
15.1 
15.1 
15.1 
15.1 
15.1 
15.1 
15.1 

15.1 

15.2 
15.2 
15.2 
15.2 

15.2 

16.2 
15.2 
15.2 
15.2 

15.2 

15.3 
15.3 

15.3 

16.3 

15.3 

).845227 
845496 
845764 
846033 
846302 
846570 
846839 
847107 
847376 
847644 
847913 
9.848181 
848449 
848717 
848986 
849254 
849522 
849790 
850058 
850325 
850593 
9.850861 
851129 
851396 
851664 
851931 
852199 
852466 
852733 
853001 
853268 
9.853535 
853802 
854069 
854336 
864603 
854870 
855137 
855404 
855671 
855938 
9.856204 
856471 
856737 
857004 
857270 
857537 
857803 
858069 
858336 
858602 
9.858868 
859134 
859400 
859666 
859932 
860198 
860464 
860730 
860995 
861261 

44.8 

44.8 

44.8 

44.8 

44.8 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.7 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.6 

44.5 

44.5 

44.5 

44.5 

44.5 

44.6 
44.5 

44.5 

44.6 
44.6 
44.5 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.4 
44.3 
44.3 
44.3 
44.3 
44.3 
44.3 
44.3 
44.3 
44.3 
44.3 

10.1547731 
154504 I 
154236: 
153967 
153698 
153430 
1531C1 
152893 
152624 
152356 
162087 
10.151819 
161551 
151283 
161014 
150746 
150478 
160210 
149942 
149675 
149407 
10.149139 
148871 
148604 
148336 
148069 
147801 
147534 
147267 
146999 
146732 
10-146465 
146198 
145931 
145664 
145397 
145130 
144863 
144596 
144329 
144062 
10-143796 
143529 
143263 
142996 
142730 
142463 
142197 
141931 
141664 
141398 
10141132 
140866 
140600 
140334 
140068 
139802 
139536 
139270 
139005 
138739 

57358 
57381 
57405 
67429 
67453 
57477 
67501 
67524 
67548 
57572 
67696 
57619 
57643 
57667 
57691 
67715! 
67738 
67762 
57786 
67810 
57833 
57857 
57881 
57904 
57928 
57952 
67976 
67999 
68023 
58047 
68070 
68094 
68118 
68141 
68166 
68189 
58212 
58236 
58260 
68283 
68307 
58330 
68354 
58378 
58401 
68425 
58449 
68472 
168496 
58519 
58543 
68567 
58590 
58614 
68637 
1 58661 
68684 
58708 
58731 
68755 
58779 

81915 
81899 
81882 
81866 
81848 
81832 
81815 
81798 
81782 
81765 
81748 
81731 
81714 
81698 
81681 
81664 
81647 
81631 
81614 
81697 
81580 
81663 
81546 
81530 
81613 
81496 
81479 
81462 
81445 
81428 
81412 
81395 
81378 
81361 
81344 
81327 
81310 
81293 
81276 
81259 
81242 
81225 
81208 
81191 
81174 
81157 
81140 
81123 
81106 
81089 
81072 
81055 
81038 
81021 
81004 
80987 
80970 
80953 
80036 
80919 
80902 

60 

69 

58 

67 

66 

55 

54 

53 

62 

61 

60 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

1 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Cotang. 

Tang. 

N. cos 

N.sine 

/ 


54 Degrees. 





































































.. . —- 

TABLE II. Log. Sines and Tangents. (3C°) Natural Sines. 57 


r 

Sine. 

D. 10" 

Cosine. 

D. 10 7 ’ 

Tang. 

D. 10" 

Cotang. N. sine. 

N. cos. 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 
58 

57 

58 

59 

60 

9.769219 

769393 

769566 

769740 

769913 

770087 

770260 

770433 

770606 

770779 

770952 

9.771125 

771298 

771470 

771643 

771815 

771987 

772159 

772331 

772503 

772676 

9.772847 

773018 

773190 

773361 

773533 

773704 

773875 

774046 

774217 

774388 

9.774558 

774729 

774899 

775070 

775240 

775410 

775580 

775750 

775920 

776090 

9.776259 

776429 

776598 

776768 

776937 

777106 

777275 

777444 

777613 

777781 

9.777950 

778119 

778287 

778455 

778624 

778792 

778960 

779128 

779295 

779463 

29.0 

28.9 

28.9 

28.9 

28.9 

28.9 

28.8 

28.8 

28.8 

28.8 

28.8 

28.8 

28.7 

28.7 

28.7 

28.7 

28.7 

28.7 

28.6 

28.6 

28.6 

28.6 

28.6 

28.6 

28.5 

28.5 

28.6 
28.5 
28.5 
28.5 
28.4 
28.4 
28.4 
28.4 
28.4 
28.4 
28.3 
28.3 
28.3 
28.3 
28.3 
28.3 
28.2 
2«.2 
28.2 
28.2 
28.2 
28.1 
28.1 
28.1 
28.1 
28.1 
28.1 
28.0 
28.0 
28.0 
28.0 
28.0 
28.0 
27.9 

9.907958 
907866 
907774 
907682 
907590 
907498 
907406 
907314 
907222 
907129 
907037 
9.906945 
906852 
906760 
906667 
906575 
906482 
906389 
906296 
906204 
906111 
9.906018 
905925 
905832 
905739 
905645 
905552 
905459 
905366 
905272 
905179 
9.905085 
904992 
904898 
904804 
904711 
904617 
904523 
904429 
904335 
904241 
9.904147 
904053 
903959 
903864 
903770 
903676 
903581 
903487 
903392 
903298 
9.903202 
903108 
903014 
902919 
902824 
902729 
902634 j 
902539 
902444 
902349 

15.3 

15.3 

15.3 

15.3 

15.3 

15.3 

15.3 

15.4 
15.4 
15.4 

15.4 

16.4 

15.4 

15.4 

16.4 

15.4 

15.4 

15.5 
15.5 

15.5 
15*6 

15.6 
15.5 

15.5 

15.6 
15.6 
15.5 

15.5 

15.6 

1 >.6 
15.6 
15.6 
15.6 
15.6 
15.6 
15.6 

15.6 
15.6 

15.7 
15.7 
15.7 
15.7 
15.7 
15.7 
15.7 

15.7 

16.7 

15.7 

15.7 

16.8 
15.8 

15.8 

16.8 
15.8 
15.8 
15.8 
15.8 

15.8 

15.9 
15.9 

9.861261 

861527 

861792 

862058 

862323 

862589 

862854 

863119 

863385 

863650 

863916 

9.864180 

864445 

864710 

864975 

865240 

865505 

865770 

866036 

866300 

866564 

9.866829 

867094 

867358 

867623 

867887 

868152 

868416 

868680 

868945 

869209 

9.869473 

869737 

870001 

870265 

870629 

870793 

871057 

871321 

871685 

871849 

9.872112 

872376 

8/2640 

872903 

873167 

873430 

873694 

873957 

874220 

874484 

9.874747 

876010 

876273 

875536 

876800 

876063 

876326 

876589 

876851 

877114 

44.3 

44.3 

44 2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.2 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.1 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

44.0 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.9 

43.8 

43.8 

43.8 

43.8 

43.8 

43.8 

43.8 

10.138739 
138473 
138208 
137942 
137677 
137411 
137146 
136881 
136615 
136360 
136085 
10.135820 
135655 
136290 
135025 
134760 
134496 
134230 
133965 
133700 
133436 
10.133171 
132906 
132642 
132377 
132113 
131848 
131584 
131320 
131056 
130791 
10.130527 
130263 
129999 
129735 
129471 
129207 
128943 
128679 
128415 
128151 
10.127888 
127624 
127360 
127097 
126833 
126570 
1263061 
126043 1 
125780 
125616 
10.125253 
124990 
124727 
124464 
124200 
123937 
123674 
123411 
123149 
122886 

68779 
68802 
68826 
68849 
68873 
68896 
68920 
58943 
168967 
58990 
69014 
:59037 
59061 
69084 
69108 
59131 
|59154 
69178 
69201 
69225 
j59248 
!59272 
59295 
69318 
69342 
59366 
59389 
59412 
69436 
69459 
69482 
69506 
59529 
59562 
59676 
59599 
59622 
59646 
69669 
69693 
59716 
69739 
59763 
69786 
59809 
59832 
69856 
59879 
59902 
59926 
69949 
59972 
59995 
60019 
60042 
60065 
60089 
60112 
60135 
60158 
60182 

80902 

'80885 

80867 

80860 

80833 

80816 

80799 

80782 

80765 

80748 

80730 

80713 

80696 

80679 

80662 

80644 

80627 

80610 

80593 

80576 

80568 

80541 

80524 

80507 

80489 

80472 

80455 

80438 

80422 

80403 

80386 

80368 

80361 

80334 

80316 

80299 

80282 

80264 

80247 

80230 

80212 

80196 

80178 

80160 

80143 

80126 

80108 

80091 

80073 

80056 

80038 

80021 

80003 

79986 

79968 

79951 

79934 

79916 

79899 

79881 

79864 

60 

69 

58 

67 

56 

56 

54 

53 

62 

51 

60 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 

1 

Sine. 


C* tang. 


Tang. 

N. cos. 

N.sine. 

# 

53 Degrees. 

































































58 Log. Sines and Tangents. (37°) Natural Sines. TABLE II. 


/ 

Sine. 

D. 10' 

' Cosine. 

| D. 10' 

' Tang. 

D. 10 

' Cotang. 

j N .sine 

. N. cos 

. 

0 

9.779483 

27.9 

27.9 

27.9 

27.9 

27.9 

27.8 

27.8 

27.8 

27.8 

27.8 

27.8 

27.7 

27.7 

27.7 

27.7 

27.7 

27.7 

27.6 

27.6 

27.6 

27.6 

27.0 

27.6 

27.5 

27.5 

27.5 

27.5 

27.6 
27.5 
27.4 
27.4 
27.4 
27.4 
27.4 
27.4 

9.902349 

15.9 
15.9 
15.9 
15.9 
15 9 
15 9 
15 9 
15.9 
15.9 

15 9 

16 0 
16 0 
16 0 
16 0 
16 0 
16 0 
16 0 
160 
16 0 
10 0 
10 1 
10 1 
10 1 
10.1 
10.1 
16 1 
16.1 
10.1 
10.1 
10.1 
10 2 
10.2 
16.2 
10.2 
10.2 
10.2 
16.2 
16.2 
16.2 
16.2 

10.3 

16.3 

10.3 

16.3 

10.3 

10.3 

16.3 

10.3 
10.3 
10.3 

10.3 

16.4 

10.4 
10.4 
10.4 
10.4 

10.4 

16.4 
16.4 
16.4 

9.877114 

43.8 
43.8 
43.8 
j 43.8 
43.8 
43.8 
43.8 
43.7 
43.7 
43 7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.7 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.6 
43.0 
43.0 
43.6 
43.0 
43.0 
43.0 
43.6 
43.0 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 
43.5 

43.5 

43.6 

43.5 

43.6 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 
43.4 

10.122886 

1 6018^ 

79864 

60 

1 

779031 

902253 

877377 

122623 

0020c 

79840 

59 

2 

779798 

902158 

877640 

122360 

00228 

79829 

58 

3 

779900 

902003 

877903 

122097 

00251 

79811 

57 

4 

780133 

901907 

878165 

121835 

| 60274 

79793 

56 

5 

780300 

90i872 

878428 

121572 

00298 

79770 

55 

6 

780407 

901770 

878091 

121309 

60321 

79758 

54 

7 

780034 

901081 

878953 

121047 

60344 

79741 

53 

8 

780801 

901585 

879216 

120784 

60367 

79723 

62 

9 

780908 

9014y0 

879478 

120522 

|60390 

79700 

|51 

10 

781134 

901394 

879741 

120259 

60414 

79688 

50 

11 

9.781301 

9.901298 

9.880003 

10.119997 

60437 

79671 

49 

12 

781408 

901202 

880265 

119735 

60400 

79658 

48 

13 

781034 

901100 

880528 

119472 

00483 

79635 

47 

’ 14 

781800 

901010 

880790 

119210 

60500 

79618 

46 

15 

781900 

900914 

881052 

118948 

60529 

79600 

45 

10 

782132 

900818 

881314 

118080 

60553 

79583 

44 

17 

782298 

900722 

881676 

118424 

60576 

79565 

43 

18 

782404 

900620 

881839 

118101 

60599 

79547 

42 

19 

782030 

900329 

882101 

117899 

00022 

79530 

41 

20 

21 

782790 

9.782951 

900433 

9.900337 

882363 

9.882025 

117637 

10.117375 

60045 

60008 

79512 

79494 

40 

39 

22 

783127 

900242 

882887 

117113 

60091 

79477 

38 

23 

783282 

900144 

883148 

116852 

60714 

79459 

37 

24 

783458 

900047 

883410 

110590 | 

60738 

79441 

36 

25 

783023 

899951 

883072 

116328 

60701 

79424 

35 

20 

27 

28 
29 

783788 

783953 

784118 

784282 

899854 

899757 

899000 

899504 

883934 

884190 

884457 

884719 

116006 

115804 

115543 
115281 

60784 

60807 

60830 

60853 

79400 

79388 

79371 

79353 

34 

33 

32 

31 

30 

784447 

899407 

884980 

115020 

60876 

79335 

30 

31 

32 

9.784012 

784770 

9.899370 

899273 

9.885242 

885503 

10.1U758 
114497 I 

oooyy 

6092-j 

79318 

79300 

29 

28 

33 

784941 

899170 

885765 

114235 

60945 

79282 

27 

34 

35 

785105 

785209 

899078 

898981 

886020 

880288 

113974 
113712 ! 

00905 

60991 

79204 

79247 

20 

26 

30 

785433 

27.3 

27.3 

27.3 

27.3 

27.3 

27.3 

27.2 

27.2 

27.2 

27.2 

898884 

886549 

113451 

61015 

79229 

24 

37 

785597 

898787 

880810 

113190 

01038 

79211 

23 

38 

39 

785701 

785925 

898089 

898592 

887072 

887333 

112928 

112067 

01001 

61084 

79193 

.9170 

22 

21 

40 

41 

780089 

9.780252 

898494 

9.898397 

887594 

9.887855 

112406 

10.112145 

61107 

61130 

79158 

79140 

20 

19 

42 

780416 

898299 

888116 

111884 

61153 

79122 

18 

43 

780579 

898202 

888377 

111623 

61176 

79105 

17 

44 

780742 

898104 

888639 

111301 

01199 

79087 

10 

45 

780900 

898000 

888900 

111100! 

61222 

79009 

15 

40 

787009 

27.2 
27.2 
27.1 
27.1 

897908 

889100 

110840 ! 

61245 

79051 

14 

47 

787232 

897810 

889421 

110579 

61208 

79033 

13 

48 

787395 

897712 

889682 

110318 

61291 

79010 

12 

49 

787557 

897014 

889943 

110057 

61314 

.8998! 

11 ’ 

50 

787 720 

27.1 

897510 

890204 

109796 :61337! 

78980I 

10 | 

51 

9.787803 

27*1 

27.1 

27.1 

27.1 

27.0 

27.0 

27.0 

27.0 

27.0 

27.0 

9.897418 

9.890405 

10.109536 61300! 

789621 

9 I 

52 

63 

788045 
788208i 

897320 

897222 

890726 

890980 

109275 

109014 

61383 

61400 

78944' 
78926 

8 

7 

; 54 

788370 

897123 

891247 

108753 i| 

61429 78908 

6 

55 

788532 

897025 

891507 

108493 

61451 

78891 

5 

66 

788094 

890920 

891708 

108232 'j 

01474' 

78873 

4 1 

57 

788850 

896828 

892028 

10.972i 

61497 

78855 

3 

58 

789018 

890729 

892289 

107711j 

01520 

78837 

2 

59 

789180 

896031 

892549 

107451| 

61543 

78819 

1 

00 

789342 

890532 

892810 

107190 j 

6156o 

78801 

0 


Cosine, t 

Sine. 


Cotang. 


Tang. 

N. cos. 

N.sine. 

t 


52 Degrees. 































































































TABLE II. Log. Bines and Tangents. (38°) Natural Sinaa. 59 


1 

Sine. 

D. 10" 

Cosine. 

D. 10" 

Tang. 

I). 10"] 

Cotang. | N. sine. 

N. cos. 


0 

9.789342 

26.9 

26.9 

26.9 

26.9 

26.9 

26.9 

26.8 

26.8 

26.8 

26.8 

26.8 

26.8 

26.7 

26.7 

26.7 

26.7 

26.7 

26.7 

26.6 

26.6 

26.6 

26.6 

26.6 

26.6 

26.6 

26.6 

26.6 

26.5 

26.6 
26.5 
26.4 
26.4 
26.4 
26.4 
26.4 
26-4 
26-4 
26-3 
263 
26-3 
26-3 
263 
26-3 
26-3 
26-2 
26-2 
26-2 
26-2 
26-2 
26-1 
26-1 
26-1 
26-1 
26-1 
26-1 
26-1 
26.1 
26.0 
26.0 
26.0 

9.896532 

16.4 

16.5 

16.5 

16.6 
16.5 

16.5 

16.6 
16.6 

16.5 

16.6 

16.5 

16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.6 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 
16.7 

16.7 

16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.8 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
16.9 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 
17.0 

9.892810 

43.4 

43.4 

43.4 

43.4 

43.4 

43.4 

43.4 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43.3 

43-2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.2 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.1 

43.0 

10.107190i 

161566 

78801 

60 

1 

789504 

896433 

8930ro 

106930 

61589 

78783 

69 

2 

789665 

896335 

893331 

106669 

61612 

78765 

58 

3 

789827 

896236 

893591 

106409! 

61635 

78747 

67 

4 

789988 

896137 

893861 

106149 

61658 

78729 

66 

5 

790149 

896038 

894111 

105889 

61681 

78711 

65 

6 

790310 

895939 

894371 

105629 

61704 

78694 

54 

7 

790471 

895840 

894632 

105368 

61726 

78676 

53 

8 

790632 

895741 

894892 

105108 

61749 

78668 

52 

9 

790793 

895641 

895152 

104848 

61772 

78640 

61 

10 

790954 

895542 

895412 

104588 

61796 

78622 

60 

11 

9.791115 

9.895443 

9.895672 

10.104328 

61818 

78604 

49 

12 

791275 

895343 

895932 

104068 

61841 

78586 

48 

13 

791436 

895244 

896192 

103808 

61864 

78668 

47 

14 

791596 

895145 

896452 

103548 

61887 

78550 

46 

15 

791757 

895045 

896712 

103288 

61909 

78532 

45 

16 

791917 

894945 

896971 

103029 

61932 

78514 

44 

17 

792077 

894846 

897231 

102769 

61956 

78496 

43 

18 

792237 

894746 

897491 

102509 

61978 

78478 

42 

19 

792397 

894646 

897761 

102249 

62001 

78460 

41 

20 

792557 

894546 

898010 

101990 

62024 

78442 

40 

21 

9.792716 

9.894446 

9.898270 

10.101730 

62046 

78424 

39 

22 

23 

792876 

793035 

894346 

894246 

898530 

898789 

101470 

101211 

62069 

62092 

78406 

78387 

38 

37 

24 

793195 

894146 

899049 

100951 

62115 

78369 

36 

26 

793354 

894046 

899308 

100692 

62138 

78351 

36 

26 

793614 

893946 

699568 

100432 

62160 

78333 

34 

27 

793673 

893846 

899827 

100173 

62183 

78315 

33 

28 

793832 

893745 

900086 

099914 

62206 

78297 

32 

29 

793991 

893645 

900346 

099654 

62229 

?8279 

31 

30 

794150 

893544 

900605 

099395 

62251 

78261 

30 

31 

9.794308 

9.893444 

9.900864 

10.099136 

62274 

78243 

29 

32 

794467 

893343 

901124 

098876 

62297 

78225 

28 

33 

794626 

893243 

901383 

098617 

62320 

78206 

27 

34 

794784 

893142 

901642 

098358 

62342 

78188 

26 

35 

794942 

893041 

901901 

098099 

62365 

78170 

25 

36 

795101 

892940 

902160 

097840 

62388 

78162 

24 

37 

795259 

892839 

902419 

097581 

62411 

78134 

23 

38 

795417 

892739 

902679 

097321 

62433 

78116 

22 

39 

795576 

892638 

902938 

097062 

62456 

78098 

21 

40 

795733 

892536 

903197 

096803 

62479 

78079 

20 

41 

9.795891 

9.892435 

9.903455 

10.096545 

62602 

78061 

19 

42 

796049 

892334 

903714 

096286 

62524 

78043 

18 

43 

796206 

892233 

903973 

096027 

62647 

78026 

17 

44 

796364 

892132 

904232 

096768 

62670 

78007 

16 

45 

796521 

892030 

904491 

095509 

62692 

77988 

16 

46 

796679 

891929 

904750 

095250 

62615 

77970 

14 

47 

796836 

891827 

905008 

094992 

62638 

77952 

13 

48 

796993 

891726 

905267 

094733 

62660 

77934 

12 

49 

797150 

891624 

905526 

094474 

62683 

77916 

11 

50 

797307 

891623 

906784 

094216 

62706 

77897 

10 

51 

9.797464 

9.891421 

9.906043 

10.093957 

62728 

77879 

9 

52 

797621 

891319 

906302 

093698 

62761 

77861 

8 

53 

797777 

891217 

906560 

093440 

62774 

77843 

7 

64 

797934 

891115 

906819 

093181 

62796 

77824 

6 

55 

798091 

891013 

907077 

092923 

62819 

77806 

6 

56 

798247 

890911 

907336 

092664 

62842 

77788 

4 

57 

798403 

890809 

907694 

092406 

62864 

77769 

3 

68 

798560 

890707 

907852 

092148 

62887 

77751 

2 

69 

798716 

890605 

908111 

091889 

62909 

77733 

1 

60 

798872 

890503 

908369 

091631 

62932 

77715 

0 


Cosine. 

Sine. 


Cotang. 


Tang. 

N. cos. 

N^ine- 

t 


51 Degrees. 


20 


















































60 Log. Sines and Tangents. (39°) Natural sines. TABLE II. 


J 

Sine. 

D. 10 

Cosine. 

D. 10" 

TangJ 

D. 10' 

Co tang. 

iN. sine 

N. cos 

1 

0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 
11 
12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 
64 

55 

56 

57 

58 

59 

60 

9.798772 

799028 

799184 

799339 

799495 

799651 

799806 

799962 

800117 

800272 

800427 

9.800582 

800737 

800892 

801047 

801201 

801356 

801511 

801665 

801819 

801973 

9.802128 

802282 

802436 

802589 

802743 

802897 

803050 

803204 

803357 

803511 

9.803664 

803817 

803970 

804123 

804276 

804428 

804581 

804734 

804886 

805039 

9.805191 

805343 

805495 

805647 

805799 

805951 

805103 

806254 

806406 

806557 

9.806709 

806860 

807011 

807163 

807314 

807465 

807615 

807766 

807917 

808067 

26.0 
26.0 
26.0 
25.9 
25.9 
25.9 
25.9 
25.9 
25.9 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.8 
25.7 
25.7 
25.7 
25.7 
25.7 
25.7 
25.6 
25.6 
25.6 
25.6 
25.6 
25.6 
25.6 
25.5 
25.5 
25.5 
25.5 
25.5 
25.5 
25-4 
25*4 
25.4 
25.4 
25.4 
25.4 
25.4 
25-3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.3 
25.2 
25.2 
25.2 
25.2 
25.2 
25.2 
25.2 
25.1 
25.1 
25.1 
25.1 

9.890503 
890400 
890-298 
890195 
890093 
889990 
889888 
889783 
889682 
889579 
889477 
9.889374 
889271 
889168 
889064 
888961 
888858 
888755 
888651 
888548 
888444 
9.888341 
888237 
888134 
888030 
887926 
887822 
887718 
887614 
887510 
887406 
9.887302 
887198 
887093 
886989 
88b885 
886780 
886676 
886571 
886466 
886362 
9.886257 
886152 
886047 
885942 
885837 
885732 
88562 7 
885522 
885416 
885311 

9.865205 
885100 
884994 
884889 
884783 
884677 
884572 
884466 
884360 
884254 

17.0 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.1 

17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 
17.2 

17.2 

17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 
17.3 

17.3 

17.4 
17.4 
17.4 
17.4 
17.4 
17.4 
17.4 
17.4 
17.4 

17.4 

17.5 
17.5 
17.5 

17.5 

17.6 

17.5 

J 7.5 

17.6 

17.5 

17.6 
17.6 
17.6 
17.6 
17.6 
17.6 
17.6 
17.6 
17.6 
17.6 
17.6 

9.903369 

908628 

903886 

900144 

900402 

909660 

909918 

910177 

910435 

910693 

910951 

9.911209 

911467 

911724 

911982 

912240 

912498 

912756 

913014 

913271 

913529 

9.913787 

914044 

914302 

914560 

914817 

915075 

915332 

915590 

915847 

916104 

9.916362 

916619 

916877 

917134 

917391 

917648 

917905 

918163 

918420 

918677 

9.918934 

919191 

919448 

919705 

919962 

920219 

920476 

920733 

920990 

921247 

9 921503 
921760 
922017 
922274 
922530 
922787 
923044 
923300 
923557 
923813 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

43.0 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.9 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.8 

42.7 

10.091631 
091372 
001114 
09J856 
090598 
090340 
090082 
089823 
089566 
089307 
089049 
10.088791 
088533 
088276 
088018 
087760 
087502 
087244 
086986 
086729 
086471 
10-086213 
085956 
085698 
085440 
085183 
084925 
084668 
084410 
084153 
083896 
10-083638 
083381 
083123 
082866 
082609 
082352 
082095 
081837 
081580 
081323 
10-081066! 
080809 
080552 
080295 
080038 
079781 
079524 
079267! 
079010' 
078753! 
10-0784971 
0782401 
077983! 
077726! 
0774701 
077213, 
076956 i 
076700 
076443 
076187! 

62932 
62956 
j629 7/ 
6300J 
63022 
63045 
63058 
63090 
63113 
63135 
63158 
93180 
63203 
63225 
63248 
63271 
63293 
63316 
63338 
63361 
63383 
63405 
63428 
63451 
63473 
63496 
63518 
6-3540 
!63563 
635.55 
6360o 
63630 
63653 
63675 
63698 
63720 
63 742 
63765 
63787 
63810 
63832 
63854 
6387 < 
63899 
63922 
63944 
6396b 
6398c 
64011 
64033 
64056 
64078 
64100 
64123 
64145 
64167 
64190 
64212 
64234 
64256 
64279 

77715 

77696 

77678 

77660 

77641 

77623 

77605 

77586 

77568 

77550 

77531 

77513 

77494 

77476 

77458 

77439 

77421 

77402 

77 384 

77366 

77347 

77329 

77310 

77292 

77273 

77255 

77236 

77218 

77199 

77181 

77162 

77144 

77125 

77107 

7 70o8 
77070 
77051 
77033 

7 7014 
76996 
76977 
76959 
76940 
76921 
76903 
76884 
7o86o 
76847 
76828 
76810 
76791 
76772 
76754 
76735 
76717 
76698 
76679 
76661 
76642 
76623 
76604 

'60 
i 59 
58 
57 
56 
55 
64 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Co tang. 


Tang. | 

N. cos. 

N.sine. 






50 Degrees. 


































































~r 

TABLE EE. 

Log. Sines 

and Tangents. (40°) Natural Sines. 

6i 


1 Sine. 

D. 10' 

Cosine. 

1). 10' 

Tang. 

D. 10' 

Cotang. 

N .sine 

N. cos 


0 

1 

2 

8 

4 

6 

6 

7 

8 
9 

10 

i 11 

1 12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 
, 25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 

51 

52 

53 
64 
66 
66 

57 

58 
69 
60 

9.80S067 

808218 

808368 

808519 

808369 

808819 

808939 

809119 

809269 

809419 

809539 

9.809718 

809838 

810017 

810167 

810316 

810465 

810614 

810763 

810912 

811031 

9.811210 

811358 

811507 

811655 

811804 

811952 

812100 

812248 

812396 

812544 

9.812692 

812840 

812988 

813135 

813283 

813430 

813578 

813725 

813872 

814019 

9.814166 

814313 

814460 

814607 

814753 

814900 

815046 

815193 

815339 

815485 

9.815631 

815778 

815924 

816059 

816215 

816361 

816507 

816662 

816798 

816943 

25.1 

25. 1 

25.1 

25.0 

25.0 

25.0 

25.0 

25.0 

25.0 

24.9 

24.9 

24.9 

24.9 

24.9 

24.9 

24.8 

24.8 

24.8 

24 8 

24> 

24.8 

24.8 

24.7 

24.7 

24.7 

24.7 

24.7 

24.7 

24.7 

24.6 

24.6 

24.6 

24.6 

24.6 

24.6 

24.6 

24.5 

24.5 

24.5 

24.6 
24.6 
24.5 
24.5 
24.4 
24.4 
24.4 
24.4 
24.4 
24.4 
24.4 
24.3 
24.3 
24.3 
24.3 
24.3 
24.3 
24.3 
24.2 
24.2 
24.2 

9.884254 
884148 
884012 
883936 
883829 
883723 
883617 
883510 
883404 
883297 
883191 
9.883084 
882977 
882871 
882764 
882657 
882550 
882443 
882336 
882229 
882121 
9.882014 
881907 
881799 
881692 
881584 
881477 
881369 
881261 
881163 
881046 
9.880938 
880830 
880722 
880613 
880505 
880397 
880289 
880180 
880072 
879963 
9.879855 
879746 
879637 
879529 
879420 
879311 
879202 
879093 
878984 
878875 
9.878766 
878656 
878547 
878438 
878328 
878219 
878109 
877999 
877890 
877780 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.7 

17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 
17.8 

17.8 

17.9 
17.9 
17.9 
17.9 
17.9 
17.9 
17.9 
17.9 
17.9 
17.9 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.0 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.1 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.2 
18.3 
18.3 
18.3 
18.3 

■9.923813 

924070 

924327 

924583 

924840 

925096 

925352 

925609 

925865 

926122 

926378 

9.926634 

926890 

927147 

927403 

927659 

927915 

928171 

928427 

928683 

928940 

9.929196 

929462 

929708 

929964 

930220 

930475 

930731 

930987 

931243 

931499 

9.931755 

932010 

932266 

932622 

932778 

933033- 

933289 

933545 

933800 

934056 

9.934311 

934567 

934823 

935078 

935333 

935589 

935844 

936100 

936355 

936610 

9.936866 

937121 

937376 

937632 

937887 

938142 

938398 

938653 

938908 

939163 

42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.7 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
*42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 
42 .-6 
42.6 
42.6 
42.6 
42.6 
42.6 
42.6 

42.5 

42.6 
42.6 
42.5 
42.5 

42.5 

42.6 
42.6 
42.6 

10.076187 

075930 

075673 

075417 

075160 

074904 

0?4648 

074391 

074135 

073878 

073622 

10.073366 

073110 

072863 

072597 

072341 

072085 

071829 

071573 

071317 

071060 

10.070804 

070548 

070292 

070036 

069780 

069525 

069269 

069013 

068757 

068501 

10.068245 

067990 

067734 

067478 

067222 

066967 

066711 

066455 

066200 

065944 

10.065689 

065433 

066177 

064922 

064667 

064411 

064156 

063900 

063645 

063390 

10.063134 

062879 

062624 

062368 

062113 

061868 

061602 

061347 

061092 

060837 

64279 

64301 

64323 

64346 

64368 

64390 

64412 

64435 

64457 

64479 

64501 

64524 

64546 

64568 

64590 

64612 

64635 

64657 

64679 

64701 

64723 

64746 

64768 

64790 

64812 

64834 

64856 

64878 

64901 

64923 

64946 

64967 

64989 

65011 

65033 

65055 

65077 

65100 

65122 

65144 

65166 

66188 

65210 

65232 

65254 

66276 

66298 

65320 

65342 

65364 

65386 

65408 

65430 

66452 

65474 

65496 

65618 

65540 

66562 

65684 

65606 

76604 

76586 

76567 

76548 

76530 

76511 

76492 

76473 

76455 

76436 

76417 

76398 

76380 

76361 

76342 

76323 

76304 

76286 

76267 

76248 

76229 

76210 

76192 

76173 

76164 

76135 

76116 

76097 

76078 

76059 

76041 

76022 

76003 

76984 

76965 

75946 

76927 

75908 

76889 

75870 

75861 

75832 

75813 

75794 

76775 

75766 

75738 

75719 

76700 

75680 

75661 

75642 

75623 

75604 

76685 

75566 

75547 

75628 

75509 

76490 

75471 

60 

59 

58 

57 

56 

55 

54 

53 

52 

61 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Cosine. | 

Sine. 


Cotang. 

| Tang. 1 ! N. cos. 

N.sine. 

/ 

49 Degrees. 
































































62 Log. Sines and Tangents. (41°) Natural Sines. TABLE II. 


/ 

j Sine. 

D. 10' 

Cosine. 

D. 10' 

Tang. 

D 10' 

Cotang. 

[ IN. sine 

N. cos 


0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 
20 

27 

28 

29 

30 

31 

32 

33 
31 
35 
30 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 
61 

52 

53 

64 

65 
56 
67 

58 

59 

60 

9.816943 
817088 
817233 
817379 
817524 
817668 
817813 
817958 
818103 
818247 
818392 
9.818536 
818681 
818825 
818969 
819113 
819257 
819401 
819645 
819689 
819832 
9.819976 
820120 
820263 
820400 
820560 
820693 
820836 
820979 
821122 
821265 
9.821407 
821650 
821693 
821835 
821977 
822120 
822262 
822404 
822546 
822688 
9.822830 
822972 
823114 
823255 
823397 
823539 
823680 
823821 
823963 
824104 
9.824245 
824386 
824627 
824668 
824808 
824949 
825090 
825230 
826371 
825511 

24.2 

24.2 

24.2 

24.2 

24.1 

24.1 

24.1 

24.1 

24.1 

24.1 

24.1 

24.0 

24.0 

24.0 

24.0 

24.0 

24.0 

24.0 

23.9 

23.9 

23.9 

23.9 

23.9 

23.9 

23.9 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.8 

23.7 

23.7 

23.7 

23.7 

23.7 

23.7 

23.7 

23.6 

23.6 

23.6 

23.6 

23.6 

23.6 

23.6 

23.6 

23.5 

23.6 
23.6 
23.5 
23.5 
23.5 
23.4 
23.4 
23.4 
23.4 
23.4 
23.4 

9.877780 
877670 
877560 
877450 
877340 
877230 
877120 
877010 
876899 
876789 
876678 
9.876668 
876457 
876347 
876236 
876125 
876014 
875904 
875793 
875682 
875571 
9.876460 
876348 
875237 
876126 
876014 
874903 
874791 
874680 
874568 
874456 
9.874344 
874232 
874121 
874009 
873896 
873784 
873672 
873660 
873448 
873335 
9 873223 
873110 
872998 
872885 
872772 
872659 
872547 
872434 
872321 
872208 
9.872095 
871981 
871868 
871755 
871641 
871528 
871414 
871301 
871187 
871073 

18.3 

18.3 

18.3 

18.3 

18.3 

18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 
18.4 

18.4 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 

18.5 
18.5 
18.5 

18.5 

18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.6 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 
18.7 

18.7 

18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.8 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 
18.9 

9.939163 
939418 
939673 
939928 
940183 
940438 
940694 
940949 
941204 
941458 
941714 
9.941968 
942223 
942478 
942733 
942988 
943243 
943498 
943752 
944007 
944262 
9.944517 
944771 
945026 
945281 
945535 
945790 
946045 
946299 
946654 
946808 
9.947063 
947318 
947672 
947826 
948081 
948336 
948690 
948844 
949099 
949353 
9.949607 
949862 
950116 
950370 
950625 
950879 
951133 
951388 
951642 
951896 
9.952150 
952405 
952659 
952913 
953167 
953421 
95367 5 
953929 
954183 
954437 

42.6 

42.5 

42.5 

42.6 
42.6 
42.6 
42.6 
42.6 

42.5 

42.6 
42.5 
42.5 
42.5 
42.5 
42 5 
42.5 

42.5 

42.6 

42.5 

42.6 
42.6 
42.5 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.4 
42.3 
42.3 
42.3 
42.3 
42.3 

10.060837 
060582 
060327 
060072 
059817 
059562 
059306 
059051 
058796 
058542 
058286 
10.058032 
057777 
057522 
057267 
057012 
056757 
056502 
056248 
055993 
055738 
10.055483 1 
055229i 
0549741 
054719 
054465 
054210 
053955 
053701 
053446 
053192 
10.052937 
052682 
052428 ! 
052174! 
0519191 
051664 
0514101 
051166 j 
050901 
050647 
10.050393 
050138 
049884 
049630 
049375 j 
049121 
048867 
048612 
048358 
048104 
10.047850i 
047695i 
047341; 
047087 
046833! 
046579 
046325 
046071 
045817 
045563 

65606 
65628 
! 65650 
65672 
65694 
65716 
65738 
65759 
65781 
65803 
65825 
65847 
65869 
65891 
65913 
65935 
‘ 65956 
65978 
66000 
66022 
66044 
66066 
66088 
66109 
66131 
66153 
66175 
! 66197 
66218 
66240 
66262 
66284 
66306 
66327 
66349 
66371 
66393 
66414 
66436 
66458 
66480 
66501 
66523 
66545 
66566 
66588 
66610 
66632 
66663 
66676 
66697 
66718 
66740 
66762 
66783 
66805 
66827 
66848 
66870 
66891 
66913 

75471 

75452 

75433 

75414 

75395 

75375 

75356 

75337 

75318 

75299 

76280 

75261 

75241 

75222 

75203 

75184 

75165 

75146 

76126 

76107 

75088 

75069 

75050 

75030 

75011 

74992 

74973 

74963 

74934 

74916 

74896 

74876 

74857 

74838 

74818 

74799 

74780 

74760 

74741 

74722 

74703 

74683 

74663 

74644 

74625 

74606 

74586 

74567 

74548 

74522 

74509 

74489 

74470 

74451 

74431 

74412 

74392 

74373 

74353 

74334 

74314 

60 

69 

58 

67 

66 

65 

64 

63 

62 

61 

60 

49 

48 

47 

46 

46 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

6 

4 

3 

2 

1 

0 


Cosine. 


Sine. 


Co tang. 


Tang. I 

N. cos. 

N.sine. 

f 


48 Degrees. 



























































TABLE II. Log. Sines and Tangents. (42°)' Natural Sines. 63 


1 

Sine. 

D. To 77 

Cosine. 

D. 10" 

Tang. 

D. 10" 

Cotang. , N. sine. 

N. cos. 

0 

1 

2 

3 

4 
6 
6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 
26 
26 

27 

28 

29 

30 
81 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 
60 

51 

52 
63 

54 

55 

50 

57 

58 
69 
60 

9.825511 
825651 
825791 
825951 
826071 
826211 
826351 
826491 
826631 
826770 
826910 
9.827049 
827189 
827328 
827467 
827606 
827745 
827884 
828023 
828162 
828301 
9.828439 
828578 
828716 
828855 
828993 
829131 
829269 
829407 
829545 
829683 
9.829821 
829959 
830097 
830234 
830372 
830509 
830646 
830784 
830921 
831058 
9.831195 
831332 
831469 
831606 
831742 
831879 
832015 
832152 
832288 
832425 
9 832561 
832697 
832833 
832969 
833106 
833241 
833377 
833512 
833648 
833783 

23.4 

23.3 

23.3 

23.3 

23.3 

23.3 

24.3 

23.3 
23.3 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.2 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.1 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
23.0 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.9 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.8 
22.7 
22.7 
22.7 
22.7 
22.7 
22.7 
22.7 
22.6 
22.6 
22.6 
22.6 
22.6 

22.6 

i 

9.871073 

870960 

870846 

870732 

870618 

870504 

870390 

870276 

870161 

870047 

8)9933 

9.869818 

869704 

869589 

869474 

869360 

869245 

869130 

869015 

838900 

868785 

9.868670 

868555 

868440 

868324 

868209 

868093 

867978 

867862 

867747 

867631 

9.867515 

867399 

867283 

867167 

867051 

866935 

866819 

866703 

866586 

866470 

9.866353 

866237 

866120 

866004 

865887 

865770 

865653 

865536 

865419 

865302 

9.865185 

865068 

864950 

864833 

864716 

864598 

864481 

864363 

864245 

864127 

19.0 

19.0 

19.0 

19.0 

19.0 

19.0 

19.0 

19.0 

19.0 

19.1 

19.1 

19.1 

19.1 

J9.1 

19.1 

19.1 

19.1 

19.1 

19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 
19.2 

19.2 

19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 
19.3 

19.3 

19.4 
19.4 
19.4 
19.4 
19.4 
19.4 
19.4 

19.4 
19.6 
19.6 
19.6 
19.6 
19.6 

19.5 

19.5 

19.6 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 
19.6 

9.954437 

954691 

954945 

956200 

955454 

955707 

955961 

956215 

956469 

956723 

956977 

9.957231 

957485 

957739 

957993 

958246 

958600 

958754 

959008 

959262 

969516 

9.959769 

960023 

960277 

960531 

960784 

961038 

961291 

961645 

961799 

962052 

9.962306 

962560 

962813 

963067 

963320 

963574 

963827 

964081 

964335 

964588 

9.964842 

965095 

965349 

965602 

965855 

966109 

966362 

966616 

966869 

967123 

9.967376 

967629 

967883 

968136 

968389 

968643 

968896 

969149 

969403 

969656 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

423 

42.3 

42.3 

42.3 

42.3 

42.3 

42-3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.3 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

10.045563 

045309 

045055 

044800 

044546 

044293 

044039 

043785 

043631 

043277 

043023 

10.042769 

042515 

042261 

042007 

041754 

041500 

041246 

040992 

040738 

040484 

10.040231 

039977 

039723 

039469 

039216 

038962 

038709 

038455 

038201 

037948 

10.037694 

037440 

037187 

036933 

036680 

036426 

036173 

036919 

035665 

035412 

10.035168 

034905 

034651 

034398 

034145 

033891 

033638 

033384 

033131 

032877 

10.032624 

032371 

032117 

031864 

031611 

031357 

031104 

030851 

030597 

030344 

i66913 
:66935 
66956 
66978 
i66999 
67021 
67043 
67064 
67086 
67107 
67129 
67151 
67172 
67194 
67215 
67237 
67268 
67280 
67301 
67323 
67344 
67366 
67387 
67409 
67430 
67462 
67473 
67495 
67516 
67538 
67559 
67680 
67602 
67623 
67645 
67666 
67688 
67709 
67730 
67752 
67773 
67796 
67816 
67837 
67859 
67880 
67901 
67923 
67944 
67965 
6798/ 
68008 
68029 
68051 
68072 
68093 
68115 
68136 
68157 
68179 
68200 

74314 

74295 

74276 

74256 

74237 

74217 

74198 

74178 

74159 

74139 

74120 

74100 

74080 

74061 

74041 

74022 

74002 

73983 

73963 

73944 

73924 

73904 

73885 

73865 

73846 

73826 

73806 

73787 

73767 

73747 

73728 

73708 

73688 

73669 

73649 

73629 

73610 
73590 
73570 
73561 
73531 

73611 
73491 
73472 
73452 
73432 
73413 
73393 
73373 
73.253 
73333 
73314 
73294 
73274 
73254 
73234 
73215 
73195 
73175 
73155 
73135 


Cosine. 

Sine. 


Co tang. 


Tang. 

I N. cos. 

N.sine. 


60 

59 

58 

57 

58 
55 
54 
53 
52 
61 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
36 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
16 
14 
13 
12 
11 
10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


47 Degrees. 
























































r— *- - ---- - - . 

64 Log. Sines and Tangents. (43°) Natural Sines. TABLE II. 


/ 

Sine. 

D. 10' 

' Cosine. 

ID. 10' 

Tang. 

[D. 10"l Cdtang. 

!jN .sine 

N. cos 


0 

9.833783 

22.6 

22.5 

22.5 

22.5 

22.5 

22.5 

22.5 

22.5 

22.5 

22.4 

22.4 

22.4 

22.4 

22.4 

22.4 

22.4 

22.4 

22.3 

22.3 

22.3 

22.3 

22.3 

22.3 

22.3 

22.2 

22.2 

22.2 

22.2 

22.2 

22.2 

22.2 

22.2 

22.1 

22.1 

22.1 

22.1 

22.1 

22.1 

22.1 

22.1 

22.0 

22.0 

22.0 

22.0 

22.0 

22.0 

22.0 

21.9 

21.9 

21.9 

21.9 

21.9 

21.9 

21.9 

21.9 

21.8 

21.8 

21.8 

21.8 

21.8 

9.864127 

19.6 

19.6 

19.7 
19.7 
19.7 
19.7 
19.7 
19.7 
19.7 

19.7 

19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 
19.8 

19.8 

19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
19.9 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.0 
20.1 
20.1 
20.1 
20.1 
20.1 
20.1 
20.1 
20.1 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
20.2 
20.3 
20.3 
20.3 
20.3 
20.3 
20.3 

9.969656 

1 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2 

42.2, 

42.2 

42.2 

42.2 

42.2 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

10.030344 

6820C 

73135 

60 

1 

2 

833919 

834054 

864010 

863892 

969909 

970162 

030091 

029838 

68221 
! 68242 

73116 

73096 

59 

58 

3 

834189 

863774 

970416 

029584 

i 68264 

73076 

57 

4 

834325 

863656 

970669 

029331 

68285 

73056 

56 

5 

834460 

863538 

970922 

029078 

68306 

73036 

55 

6 

834595 

863419 

971175 

028825 

68327 

73016 

54 

7 

831730 

863301 

971429 

028571 

68349 

72996 

63 

8 

834865 

863183 

971682 

028318 

68370 

72976 

52 

9 

834999 

863064 

971935 

028065 

68391 

72957 

61 

10 

835134 

862946 

972188 

027812 

68412 

72937 

60 i 

11 

9.835269 

9.862827 

9.972441 

10.027559 

68434 

72917 

49 

12 

835403 

862709 

972694 

027306 

68455172897 

48 

13 

835538 

862590 

972948 

027052 

6847672877 

47 

14 

835672 

862471 

973201 

026799 

68497 

72857 

46 

15 

835807 

862353 

973454 

026546 

68518 72837 

45 

10 

835941 

862234 

973707 

026293 

68539172817 

44 

17 

836075 

862115 

973960 

026040 

685611/2797 

43 

18 

836209 

861996 

974213 

025787 

68582|72777 

42 

19 

836343 

861877 

974466 

025534 

68603172757 

41 

20 

836477 

861758 

974719 

025281 

68624(72737 

10 

21 

9.836611 

9.861638 

9.974973 

10.025027 

68645 72717 

39 

22 

836745 

861519 

975226 

024774 

68666 72697 

38 

23 

836878 

861400 

975479 

024521 

6868872677 

37 

24 

837012 

861280 

975732 

024268 

68709 72657 

36 

25 

837146 

861161 

976985 

024015 

6873072637 

35 

26 

837279 

861041 

976238 

023762 

6875172617 

34 

27 

837412 

860922 

976491 

023509 

6877272597 

33 

28 

837546 

860802 

976744 

023256 

68793 72577 

32 

29 

837679 

860682 

976997 

023003 

68814I72557 

31 

30 

837812 

860562 

977250 

022750 

68835172537 

30 

31 

9.837945 

9.860442 

9.977503 

10.022497 

68857 

72517 

29 

32 

838078 

860322 

977756 

022244 

68878 72497 

28 

33 

838211 

860202 

978009 

021991 

6889972477 

27 

34 

838344 

860082 

978262 

021738 

68920 72457 

26 

35 

838477 

859962 

978515 

021485 

68941 

72437 

25 

36 

838610 

859842 

978768 

021232 

68962 

72417 

24 

37 

838742 

859721 

979021 

020979 

68983 

72397 

23 

38 

838875 

859601 

979274 

020726 

69004 

72377 

22 

39 

839007 

859480 

979527 

020473 

69025 

72357 

21 

40 

839140 

859360 

979780 

020220 

69046 

72337 

20 

41 

9.839272 

9 859239 

9.980033 

10.019967 

69067 

72317 

19 

42 

839404 

859119 

980286 

019714 

69088 

72297 

18 

43 

839536 

858998 

980538 

019462 

69109 

72277 

17 

44 

839668 

858877 

980791 

019209 1 

69130 

72257 

16 

45 

839800 

858756 

981044 

018956 

69151 72236 

15 

46 

839932 

858635 

981297 

018703 

6917272216 

14 

47 

840064 

858514 

981550 

018450 

6919372196 

13 

48 

840196 

858393 

981803 

018197 

69214:72176 

12 

49 

840328 

858272 

982056 

017944 

69235 72156 

11 

50 

840459 

858151 

982309 

017691 

69256 

72136 

10 

51 

9.840591 

9.858029 

9.982562 

10.017438 

69277 

72116 

9 

52 

840722 

857908 

982814 

017186 

69298 

72095 

8 

53 

840854 

857786 

983067 

016933 

69319 

72075 

7 

54 

840985 

857665 

933320 

016680 

69340 

72055 

6 

55 

841116 

857543 

983573 

016427 

69361 

72035 

6 

56 

67 

841247 

841378 

857422 

857300 

983826 

984079 

016174 

015924 

69382 

69403 

72015 

71995 

4 

3 

58 

841609 

857178 

984331 

015669 

69424 

71974 

2 

59 

841640 

857056 

984584 

015416 

69445 

71954 

1 

60 

841771 

856934 

984837 

615163 I 

69466 

71934 

0 


(' sine. 


Sine. 


Cotang. 


Tang. i 

N. cos. 

N.sine. 

f “ 


46 Degrees. 

.. - ■ . - ■ ■ ■ ■ .4 













































































j TABLE II. Lojj. Sines and Tangents. (44°) Natural Sines. 65 

1 ' 

Sine.. . 

D. lo" 

Cosine.. 

D, 10" 

Tang. 

D. 10" 

Cotang. 

IN. sine 

N. cos. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

i 9 

‘ 10 
11 
12 

13 

14 

15 

i 16 

17 

18 

10 
20 
21 
22 
23 
j 24 

3 25 
26 
j 27 

1 28 
| 20 
| 30 

31 

32 

33 

34 
j 35 

36 

37 

38 
30 

40 

41 

, 42 
! 43 
44 
! 45 

46 

47 

48 
40 
60 
51 
62 

, 53 

64 

65 

66 
67 
58 
60 
60 

0.841/71 
841002 
842033 
842163 
842294 
842424 
842555 
842685 
842815 
842946 
843076 
9.843206 
843336 
843466 
843595 
843/25 
843855 
843084 
844114 
844243 
844372 
9.844502 
844631 
844760 
844880 
845018 
845147 
845276 
845405 
845533 
845662 
9.845790 
845010 
846047 
846175 
846304 
846432 
846560 
846688 
846816 
846044 
9.847071 
847190 
847327 
847454 
847582 
847700 
847836 
847064 
848091 
848218 
9.848345 
848472 
848500 
848726 
848852 
8489/0 
849106 
840232 
840350 
840485 

21.8 

21.8 

21.8 

21.7 

21.7 

21.7 

21.7 

21.7 

21.7 

21.7 

21.7 

21.6 

21.6 

21.6 

21.6 

21.6 

21.6 

21.6 

21.5 

21.5 

21.5 

21.5 

21.5 

21.5 

21.5 

21.5 

21.5 

21.4 

21.4 

21.4 

21.4 

21.4 

21.4 

21.4 

21.4 

21.4 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.3 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.2 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 

21.1 

0.856934 
856812 
856690 
856568 
856446 
856323 
856201 
856078 
855056 
855833 
855711 
9.855588 
855465 
855342 
855210 
855096 
8549,3 
854850 
854727 
854603 
854480 
9.854356 
854233 
854100 
853986 
853862 
853738 
853614 
853490 
853366 
853242 
9.853118 
852094 
852869 
852745 
852620 
852406 
852371 
852247 
852122 
851007 
9.851872 
851747 
851622 
851407 
861372 
851246 
851121 
850906 
850870 
850746 
9.850610 
850493 
850368 
850242 
850116 
840000 
840864 
849738 
849611 
840485 

20.3 

20.3 

20.4 
20.4 
20.4 
20.4 
20.4 
20.4 
20.4 

20.4 

20.5 

20.5 

20.6 
20.5 
20.5 
20.5 
20.5 

20.5 

20.6 
20.6 
20.6 
20.6 
20.6 
20.6 
20.6 
20.6 
20.6 
20.7 
20.7 
20.7 
20.7 
20.7 
20.7 
20.7 
20.7 

20.7 

20.8 
20.8 
20.8 
20.8 
20.8 
20.8 
20.8 
20.8 
20.9 
20.9 
20.9 
20.9 
20.9 
20.9 
20.9 
20.9 
21.0 
21.0 
21.0 
21.0 
21.0 
21.0 
21.0 
21.0 

9.984837 

985090 

985343 

985596 

985848 

986101 

986354 

986607 

986860 

987112 

987365 

9.987618 

987871 

988123 

988376 

988629 

988882 

980134 

980387 

980640 

980803 

9.990145 

900398 

990651 

900003 

901156 

901400 

901662 

991914 

902167 

992420 

9.992672 

992925 

993178 

993430 

993683 

903936 

994189 

994441 

994604 

994947 

9.905199 

995452 

995705 

995957 

906210 

906463 

906715 

906068 

997221 

997473 

9.997726 

997979 

908231 

908484 

908737 

908989 

900242 

990495 

990748 

10.000000 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

42.1 

10.015163 
014910 
014657 
014404 
014152 
013899 
013646 
013393 
013140 
012888 
012635 
10.012382 
012129 
011877 
011624 
011371 
011118 
010866 
010613 
010360 
010107 
10.009855 
000602 
000340 
000007 
008844 
008501 
008338 
008086 
007833 
007580 
10-007328 
00,0/5 
006822 
006570 
006317 
006064 
005811 
005550 
005306 
005053 
10-004801 
004548 
004295 
004043 
003790 
003537 
003285 
003032| 
002779 
002527 
10-002274 
002021 
001769 
001516 
001263 
001011 
000758 
000505 
000253 
000000 

69466 
69487 
69508 
60529 
69549 
69570 
69591 
69612 
j69633 
i69654 
!69675 
60696 
[69717 
169737 
60758 
69779 
69800 
69821 
69842 
69862 
60883 
69904 
69925 
69946 
60966 
60987 
70008 
70029 
70040 
70070 
70091 
70112 
70132 
70153 
70174 
70195 
70215 
70236 
70257 
70277 
70298 
70319 
70330 
70360 
70381 
70401 
70422 
70443 
70463 
70484 
70505 
70526 
70546 
70567 
70587 
70608 
70628 
70649 
70670 
70690 
70711 

71934 

71914 

71894 

71873 

71853 

71833 

71813 

71792 

71772 

71752 

71732 

71711 

71691 

71671 

71650 

71630 

71610 

71500 

71569 

71549 

71529 

71508 

71488 

71468 

71447 

71427 

71407 

71386 

71366 

71345 

71325 

71305 

71284 

71264 

71243 

71223 

71203 

71182 

71162 

71141 

71121 

71100 

71080 

71059 

71039 

71010 

70998 

70978 

70957 

70037 ’ 

70916 

70896 

70875 

70855 

70834 

70813 

70793 

70772 

70752 

70731 

70711 

60 

50 

58 

57 

56 

55 1 
54 | 
53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 ‘ 

6 

6 

4 

3 

2 

1 

0 

1 Cosine. 


Sine. 


Co tang. 


Tang. 

N. cos. 

N.sine. 

t 

45 Degrees. 































































66 


LOGARITHMS 

' ; 



TABLE III. 




LOGARITHMS OF NUMBERS. 



From I to 200, 



INCLUDING 

TWELVE DECIMAL 

PLACES 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

000000 000000 

41 

612783 856720 

81 

908485 018879 

2 

301029 996664 

42 

623249 290398 

82 

913813 852384 

3 

477121 254720 

43 

633468 455580 

83 

919078 092376 

4 

602059 991328 

44 

643452 676486 

84 

924279 286062 

5 

698970 004336 

1 46 

653212 513775 

85 

929418 925714 

6 

778161 250384 

46 

662767 831682 

86 

934498 451244 

7 

845098 040014 

47 

672097 857926 

87 

939519 252619 

8 

903089 986992 

48 

681241 237376 

88 

944482 672160 

9 

954242 509439 

49 

690196 080028 

89 

949390 006645 

10 

Same as to 1. 

60 

Same as to 5. 

90 

Same as to 9. 

11 

041392 686158 

61 

707570 176098 

91 

959041 392321 

12 

079181 246048 

52 

716003 343635 

92 

963787 827346 

13 

113943 352307 

53 

724275 869601 

93 

968482 948554 

14 

146128 035678 

54 

732393 759823 

94 

973127 853600 

16 

176091 259056 

65 

740362 689494 

95 

977723 605889 

16 

204119 982656 

56 

748188 027008 

96 

982271 233040 

17 

230448 921378 

57 

755874 855672 

97 

986771 734266 

18 

255272 605103 

68 

763427 993563 

98 

991226 076692 

19 

278753 600953 

59 

770852 011642 

99 

995635 194598 

20 

Same as to 2. 

60 

Same as to 6 

100 

Same as to 10, 

21 

322219 2947 

SI 

785329 835011 

101 

004321 373783 

22 

342422 680822 

62 

792391 699498 

102 

008600 171762 

23 

361727 836018 

63 

799340 549453 

103 

012837 224705 

24 

380211 241712 

64 

806179 973984 

104 

017033 339299 

26 

397940 008672 

66 

812913 356643 

105 

021189 299070 

26 

414973 347971 

66 

819543 935542 

106 

025305 865265 

27 

431363 764159 

67 

826074 802701 

107 

029383 777685 

28 

447158 031342 

68 

832508 912706 

108 

033423 755487 

29 

462397 997899 

69 

838849 090737 

109 

037426 497941 

30 

Some as to 3. 

70 

Same as to 7. 

110 

Same as to 11. 

31 

491361 693834 

71 

851258 348719 

111 

045322 978787 

32 

605149 978320 

72 

857332 496431 

112 

049218 022670 

33 

618513 939878 

73 

863322 860120 

113 

053078 443483 

34 

631478 917042 

74 

869231 719731 

114 

056904 851336 

36 

544068 044350 

75 

875061 263392 

116 

060697 840354 

36 

656302 500767 

76 

880813 592281 

116 

064457 989227 

37 

568201 724067 

77 

886490 726172 

117 

068185 861746 

38 

679783 596617 

78 

892094 602690 

118 

071882 007306 

39 

591064 607026 

79 

897627 091290 

119 

075546 961393 

40 

L=— 

Same as to 4. 

80 

Same as to 8. 

120 

Same as to 12. 

1 
































OF NUMBERS. 


67 

N. 

Log. 

N. 

Log. 

N. 

Log 

121 

082786 370316 

148 

170261 715395 

175 

243038 048686 

122 

086369 830676 

149 

173186 268412 

176 

245512 667814 

123 

089906 111439 

150 

176091 259066 

177 

241973 266362 

124 

093421 685162 

151 

178976 947293 

178 

250420 002309 

125 

096910 013008 

162 

181843 687945 

179 

262863 030980 

126 

100370 645118 

153 

184691 430818 

180 

255272 605103 

127 

103803 720956 

1 154 

187620 720836 

181 

257678 674869 

128 

107209 969648 

j 155 

190331 698170 

182 

260071 387985 

129 

110589 710299 

1 156 

193124 688364 

183 

262461 089730 

130 

Same as to 13. 

167 

195899 662409 

184 

264817 823010 

131 

117271 295656 

158 

198657 086954 

185 

267171 728403 

132 

120573 931206 

159 

201397 124320 

186 

269512 944218 

133 

123851 640967 

160 

204119 982666 

187 

271841 606536 

134 

127104 798365 

161 

206826 876032 

188 

274157 849264 

135 

130333 768495 

162 

209515 014543 

189 

276461 804173 

136 

133538 908370 

163 

212187 604404 

190 

278753 600953 

137 

136720 667166 

164 

214843 848048 

191 

281033 367248 

138 

139879 086401 

165 

217483 944214 

192 

283301 228704 

139 

143014 800254 

166 

220108 088040 

193 

285557 309008 

140 

146128 035678 

167 

222716 471148 

194 

287801 729930 

141 

149219 112655 

168 

225309 281726 

195 

290034 611362 

142 

162288 344383 

169 

227886 704614 

196 

292256 071356 

143 

165336 037466 

170 

230448 921378 

197 

294466 226162 

144 

168362 492095 

171 

232996 110392 

198 

296666 190262 

145 

161368 002235 

172 

236528 446908 

199 

298853 076410 

146 

164352 855784 

173 

238046 103129 



147 

167317 334748 

174 

240549 248283 



LOGARITHMS 

OF 

THE PRIME NUMBERS 


From 

200 to 1543, 



INCLUDING TWELVE DECIMAL PLACES. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

201 

303196 057420 

277 

442479 769064 

379 

678639 209968 

203 

307496 037913 

281 

448706 319905 ■ 

383 

683198 773968 

207 

315970 345457 

283 

451786 435524 

389 

689949 601326 

209 

320146 286111 

293 

466867 620364 

397 

698790 506763 

211 

324282 455298 

307 

487138 376477 

401 

603144 372620 

1 

' 223 

348304 863048 

311 

492760 389027 

409 

611723 308007 

227 

356025 857193 

313 

495544 337646 

419 

622214 022966 

229 

359835 482340 

317 

601059 262218 

421 

624282 095836 

233 

367355 921026 

331 

619827 993776 

431 

634477 270161 

239 

378397 900948 

337 

627629 900871 

433 

636487 896363 

241 

382017 042575 

347 

640329 474791 

439 

642424 620242 

251 

399673 721481 

349 

642825 426959 

443 

646403 726223 

257 

409933 123331 

353 

f47774 705388 

449 

652246 341003 

263 

419965 748490 

359 

665094 448678 

457 

659916 200070 

269 

429762 280002 

367 

664666 064262 

461 

663700 925390 

271 

432969 290874 

373 

571708 831809 

463 

666580 991018 














































| 68 LOGARITHMS 


N. 

Log. 

N. 

Log. 

S. 

Log. 

457 

6o9Jlo o8i)5;,6 

821 

914343 15/119 

1171 

068556 895072 

479 

680335 513414 

823 

915399 835212 

1181 

0/2249 807613 

487 

68.628 931215 

827 

917605 509553 

1187 

074450 718955 

491 

691081 492123 

829 

918554 530550 

1193 

076640 443670 

499 

6 J 0 IOJ 645623 

839 

923761 960829 

1201 

0.9543 007385 

603 

701567 986056 

853 

930949 031168 

1213 

0838S0 800845 

609 

706717 782337 

857 

932980 821923 

1217 

085290 678210 

621 

716837 723300 

859 

933993 163331 

1223 

087426 458017 

623 

718501 688867 

863 

936010 795715 

1229 

089551 882866 

641 

733197 2oo107 

877 

942999 693356 

1231 

090258 052912 

647 

737987 326333 

881 

944975 908412 

1237 

092369 699609 

667 

745855 196174 

883 

945960 703578 

1249 

096562 438356 

653 

750508 394861 

887 

947923 619632 

1269 

100026 729204 

669 

755112 26( 39 3 

907 

957607 287060 

1277 

106190 896808 

671 

756636 108243 

911 

959518 376973 

1279 

106870 642460 

677 

761175 813156 

919 

963315 611386 

1283 

108226 656362 

687 

768638 101248 

929 

968015 713994 

1-.89 

110252 917337 

593 

773064 693364 

937 

1 971739 590838 

1291 

110926 242517 

699 

777426 822389 

941 

973589 623427 

1297 

112939 986066 

601 

778874 472002 

947 

976349 979003 

1301 

114277 296540 

607 

783138 6910 5 

963 

979092 900638 

1303 

114944 415712 

613 

787460 474618 

967 

985426 474083 

1307 

116275 687564 

617 

790285 164033 

971 

987219 229908 

1319 

120244 795568 

619 

791690 649020 

977 

989894 663719 

1321 

120902 817604 

631 

800029 359244 

983 

992653 617832 

1327 

122870 922849 

641 

806858 029519 

991 

996073 654485 

1361 

133858 125188 

643 

808210 972924 

997 

998695 158312 

1367 

136768 614554 

647 

810904 280669 

1009 

003891 166237 

1373 

137670 537223 

653 

814913 181276 

1013 

005609 445360 

1381 

140193 678544 

659 

818886 414594 

1019 

008174 184006 

1399 

145817 714122 

661 

810201 459486 

1021 

009025 742087 

1409 

148910 994096 

673 

828015 064224 

1031 

013258 665284 

1423 

163204 896557 

677 

830588 668685 

1033 

014100 321520 

1427 

154424 012366 

683 

834420 703682 

1039 

016615 647557 

1429 

155032 228774 

691 

839478 047374 

1049 

020775 488194 

1433 

156246 402184 

701 

845718 017967 

1061 

021602 716028 

1439 

158060 793919 

709 

850646 235183 

1061 

026715 383901 

1447 

160468 631109 

719 

856728 890383 

1063 

026533 264523 

1451 

161667 412427 

727 

861534 410859 

1069 

028977 705209 

1463 

162265 614286 

733 

865103 974742 

1087 

036229 544086 

1459 

164055 291883 

739 

868644 4S8395 

1091 

037824 750588 

1471 

167612 672629 

743 

870988 813761 

1093 

038620 161950 

1481 

170555 058512 

761 

855639 937004 

1097 

040206 627575 

1483 

171141 151014 

757 

879095 879500 1 

1103 

042595 512440 

1487 

172310 968489 

761 

881384 656771 1 
• 

1109 

044931 646119 

1489 

172894 731332 

769 

885926 339801 

1117 

048053 173116 

1493 

174059 807708 

773 

888179 493918 

1123 

050379 756261 

1499 

175801 632866 

787 

895974 732359 

1129 

052693 941925 

1511 

179264 464329 

797 

901458 321396 

1151 

061075 323630 

1523 

182699 903324 

809 

907948 521612 

1153 

061829 307295 

1531 

184975 190807 

811 

909020 854211 

1163 

065679 714728 

1643 

188365 926063 



























































OF NUMBERS. 


69 


AUXILIARY LOGARITHMS. 


N. 

Log. 

X. 

Log. 

1.009 

003891156237 


1.0009 

000390689248 

1.008 

003460532110 


1.0008 

000347296684 

1.007 

003029470554 


1.0007 

000303899784 

1.005 

002598080685 


1.0006 

000260498647 

1.005 

002166051766 

A 

1.0005 

000217092970 

1.004 

001733712775 


1.0004 

000173683057 

1.003 

001300933020 


1.0003 

000130268804 

1.002 

000867721529 


1.0002 

000086850211 

1.001 

000434077479 J 


1.0001 

000043427277 


>B 


N. 


1.0000001 

1.00000001 

1.000000001 

1.0000000001 


Log. 

000000043429 

in) 

000000004343 

(o) 

000000000434 

(P) 

000000000043 

(q) 


X. 

Log. 

II *. 

Log. 

1.00009 

000039083266 

1.000009 

000003908628 

1.00008 

000034740691 

1.000008 

000003474338 

1.00007 

000030398072 

1.000007 

000003040047 

1.00006 

000026055410 

1.000006 

000002605766 

1.00005 

000021712704 

1.000005 

000002171464 

1.00004 

000017371430 

1.000004 

000001737173 

1.00003 

000013028638 

1.000003 

000001302880 

1.00002 

000008685802 

1.000002 

000000868587 

1.00001 

000004342923 

1.000001 

000000434294 


m=0.4342944819 log. —1.637784298. 

By the preceding tables — and the auxiliaries A, B, and 
C, we can find the logarithm of any number, true to at least 
ten decimal places. 

But some may prefer to use the following direct formula, 
which may be found in any of the standard works onalgebm: 

Log. ( 2+1 )=log.z+0.8685889638/"_L ^ 

\2z+l/ 

The result will be true to twelve decimal places, if z be 
over 2000. 

The log. of composite numbers can be determined by the 
combination of logarithms, already in the table, and the prime 
numbers from the formula. 

Thus, the number 3083 is a prime number, find its loga¬ 
rithm. 

We first find the log. of the number 3082. By factoring, 

[i we discover that this is the product of 46 into 67. 









































































\ 


70 


NUMBERS. 


Log. 46, 1.6627578316 

Log. 67, 1.8260748027 


Log. 3082 3.4888326343 


Log. 3083=3.4888326343-} 


0.8685889638 

6165 


NUMBERS AND THEIR LOGARITHMS, 

OFTEN USED IN COMPUTATIONS. 

Circumference of a circle to dia. 1 ) Log. 

Surface of a sphere to diameter it =3.14159265 0.4971499 
Area of a circle to radius 1 ) 

Area of a circle to diameter 1 = .7853982 —1.8950899 

Capacity of a sphere to diameter 1 = .5235988 —1.7189986 
Capacity of a sphere to radius 1 =4.1887902 0.6220886 

Arc of any circle equal to the radius =57°29578 1.7581226 

Arc equal to radius expressed in sec. =206264"8 5.3144251 

Length of a degree, (radius unity) = .01745329 —2.2418773 

12 hours expressed in seconds, = 43200 4.6354837 

Complement of the same, =0.00002315 —5.3615163 
360 degrees expressed in seconds, = 1296000 6.1126050 


A gallon of distilled water, when the temperature is 62° 
Fahrenheit, and Barometer 30 inches, is 277. r Vo 4 * cubic 
inches. 


«/277.274= 16.651542 nearly. 


4 


277.274 


.775398 


= 18.78925284 


4 


282. 


.785398 


= 18.948708. 


J 231 =15.198684. 
J 282 =16.792855. 


The French Metre=3.2808992, English feet linear mea¬ 
sure, =39.3707904 inches, the length of a pendulum vi¬ 
brating seconds. 



















TRAVERSE TABLE. 




71 


a> 

o 

a 

5 

% Deo. 

% Deg. 

% Deg. 

.2 

A 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1. 00 

0. 00 

1. 00 

0. 01 

1. 00 

0. 01 

2 

2. 00 

0. 01 

2. 00 

0. 02 

2. 00 

0. 03 

3 

3. 00 

0. 01 

3. 00 

0. 03 

3. 00 

0. 04 < 

4 

4. 00 

0. 02 

4. 00 

0. 03 

4. 00 

0. 05 

5 

5. 00 

0. 02 

5. 00 

0. 04 

5. 00 

0. 07 

0 

6. 00 

0. 03 

6. 00 

0. 05 

6. 00 

0. 08 

7 

7. 00 

0. 03 

7. 00 

0. 06 

7. 00 

0. 09 

8 

8. 00 

0. 03 

8. 00 

0. 07 

8. 00 

0. 10 

9 

9. 00 

0. 04 

9. 00 

0. 08 

9. 00 

0. 12 

10 

10. 00 

0. 04 

10. 00 

0. 09 

10. 00 

0. 13 

11 

11. 00 

0. 05 

11. 00 

0. 10 

11. 00 

0. 14 

12 

12. 00 

0. 05 

12. 00 

0. 10 

12. 00 

0. 16 

13 

13. 00 

0. 06 

13. 00 

0. 11 

13. 00 

0. 17 

14 

14. 00 

0. 06 

14. 00 

0. 12 

14. 00 

0. 18 

15 

15. 00 

0. 07 

15. 00 

0. 13 

15. 00 

0. 20 

16 

16. 00 

0. 07 

16. 00 

0. 14 

16. 00 

0. 21 

17 

17. 00 

0. 07 

17. 00 

0. 15 

17. 00 

0. 22 

18 

18. 00 

0. 08 

18. 00 

0. 16 

18. 00 

0. 24 

19 

19. 00 

0. 08 

19. 00 

0. 17 

19. 00 

0. 25 

20 

20. 00 

0. 09 

20. 00 

0. 17 

20. 00 

0. 26 

21 

21. 00 

0. 09 

21. 00 

0. 18 

21. 00 

0. 27 

22 

22. 00 

0. 10 

22. 00 

0. 19 

22. 00 

0. 29 

23 

23. 00 

0. 10 

23. 00 

0. 20 

23. 00 

0. 30 

24 

24. 00 

0. 10 

24. 00 

0. 21 

24. 00 

0. 31 

25 

25. 00 

0. 11 

25. 00 

0. 22 

25. 00 

0. 33 

26 

26. 00 

0. 11 

26. 00 

0. 23 

26. 00 

0. 34 

27 

27. 00 

0. 12 

27. 00 

0. 24 

27. 00 

0. 35 

28 

28. 00 

0. 12 

28. 00 

0. 24 

28, 00 

0. 37 

29 

29. 00 

0. 13 

29. 00 

0. 25 

29. 00 

0. 38 

30 

30. 00 

0. 13 

30. 00 

0. 26 

30. 00 

0. 39 

35 

35. 00 

0. 15 

35. 00 

0. 31 

35. 00 

0. 46 

40 

40. 00 

0. 17 

40. 00 

0. 35 

40, 00 

0. 52 

45 

45. 00 

0. 20 

45. 00 

0. 39 

45. 00 

0. 59 

50 

50. 00 

0. 22 

50. 00 

0. 44 

50. 00 

0. 65 

55 

55. 00 

0. 24 

55. 00 

0. 48 

55. 00 

0. 72 

60 

60. 00 

0. 26 

60. 00 

0. 52 

59. 99 

0. 79 

65 

65. 00 

0. 28 

65. 00 

0. 57 

64. 99 

0. 85 

70 

70. 00 

0. 31 

70. 00 

0. 61 

69. 99 

0. 92 

75 

75. 00 

0. 33 

75. 00 

0. 65 

74. 99 

0. 98 

80 

80. 00 

0. 35 

80. 00 

0. 70 

79, 99 

1. 05 

85 

85. 00 

0. 37 

85. 00 

0. 74 

84. 99 

1. 11 

90 

90. 00 

0. 39 

90. 00 

0. 79 

89. 99 

1. 18 

95 

95. 00 

0. 41 

95. 00 

0. 83 

94. 99 

1. 24 

100 

100. 00 

0. 44 

100. 00 

0. 87 

99. 99 

1. 31 

O 

C* 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

2 

.2 

A 

89% Deo. 

89% Deg. 

89% Deg. 


r 





















































72 TRAVERSE TABLE. 


Distance. 

1 Deg. 

1% Deg. 

1% Deg. 

1% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. j 

1 

Dep. 

1 

1. 

00 

0. 

02 

1. 

00 

0. 

02 

1. 

00 

0. 

03 

1. 

00 

0. 

03 

2 

2. 

00 

0. 

03 

2. 

00 

0. 

04 

2. 

00 

0. 

05 

2. 

CO 

0. 

06 

3 

3. 

00 

0. 

05 

3. 

00 

0. 

07 

3. 

00 

0. 

08 

3. 

00 

0. 

09 

4 

4. 

00 

0. 

07 

4. 

00 

0. 

09 

4. 

00 

0. 

10 

4. 

00 

0. 

12 

5 

5. 

00 

0. 

09 

5. 

00 

0. 

11 

5. 

00 

0. 

13 

5. 

CO 

0. 

15 

6 

G. 

00 

0. 

10 

6. 

00 

0. 

13 

6. 

00 

0. 

16 

6. 

00 

0. 

18 

7 

7. 

00 

0. 

12 

7. 

00 

0. 

15 

7. 

00 

0. 

18 

7. 

00 

0. 

21 

8 

8. 

00 

0. 

14 

8. 

00 

0. 

17 

8. 

00 

0. 

21 

8. 

00 

0. 

25 

9 

9. 

00 

0. 

16 

9. 

00 

0. 

20 

9. 

00 

0. 

24 

9. 

00 

0. 

28 

10 

10. 

00. 

0. 

17 

10. 

00 

0. 

22 

10. 

00 

0. 

26 

10. 

00 

0. 

31 

11 

11. 

00 

0. 

19 

11. 

00 

0. 

24 

11. 

00 

0. 

28 

10. 

99 

0. 

34 

12 

12. 

00 

0. 

21 

12. 

00 

0. 

26 

12. 

00 

0. 

31 

11. 

99 

0. 

37 

13 

13. 

00 

0. 

23 

13. 

00 

0. 

28 

13. 

00 

0. 

34 

12. 

99 

0. 

40 

14 

14. 

00 

0. 

24 

14. 

00 

0. 

31 

14. 

00 

0. 

37 

13. 

99 

0. 

43 

15 

15. 

00 

0. 

26 

15. 

00 

0. 

33 

14. 

99 

0. 

39 

14. 

99 

0. 

46 

18 

1G. 

00 

0. 

28 

16. 

00 

0. 

35 

15. 

99 

0. 

42 

15. 

99 

0. 

49 

17 

17. 

00 

0. 

30 

17. 

00 

0. 

37 

16. 

99 

0. 

45 

16. 

99 

0. 

52 

13 

18. 

00 

0. 

31 

18. 

00 

0. 

39 

17. 

99 

0. 

47 

17. 

99 

0. 

55 

19 

19. 

00 

0. 

33 

19. 

00 

0. 

41 

18. 

99 

0. 

50 

18. 

99 

0. 

58 

20 

20. 

00 

0. 

35 

20. 

00 

0. 

44 

19. 

99 

0. 

52 

19. 

99 

0. 

61 

21 

21. 

00 

0. 

37 

21. 

00 

0. 

46 

20. 

99 

0. 

55 

20. 

99 

0. 

64 

22 

22. 

00 

0. 

38 

21. 

99 

0. 

48 

21. 

99 

0. 

58 

21. 

99 

0. 

67 

23 

23. 

00 

0. 

40 

22. 

99 

0. 

50 

22. 

99 

0. 

60 

22, 

99 

0. 

70 

24 

24. 

00 

e. 

42 

23. 

99 

0. 

52 

23. 

99 

0. 

63 

23. 

99 

0. 

73 

25 

25. 

00 

o. 

44 

24. 

99 

0. 

55 

24. 

99 

0. 

65 

24. 

99 

0. 

76 

26 

2G. 

00 

o. 

45 

25. 

99 

0. 

57 

25. 

99 

0. 

68 

25. 

99 

0. 

79 

27 

27. 

00 

o. 

47 

26. 

99 

0. 

59 

26. 

99 

0. 

71 

26. 

99 

0. 

83 

23 

28. 

00 

o. 

49 

27- 

99 

0. 

61 

27. 

99 

0. 

73 

27. 

99 

0. 

86 

29 

29. 

00 

o. 

51 

28. 

99 

0. 

63 

28. 

99 

0. 

76 

28. 

99 

0 . 

89 

30 

30. 

00 

0. 

52 

29. 

99 

0. 

65 

29. 

99 

0. 

79 

29. 

99 

0 . 

92 

35 

34. 

99 

o. 

61 

34. 

99 

0. 

76 

34. 

99 

0. 

92 

34. 

98 

1. 

07 

40 

39. 

99 

o. 

70 

39. 

99 

0. 

87 

39. 

99 

1. 

05 

39. 

98 

1. 

22 

45 

44. 

99 

0. 

79 

44. 

99 

0. 

98 

44. 

99 

1. 

18 

44. 

98 

1. 

37 

50 

49. 

99 

o. 

87 

49. 

99 

1. 

09 

49. 

98 

1. 

31 

49. 

98 

1. 

53 

55 

54. 

99 

0. 

96 

54. 

99 

1. 

20 

54. 

98 

1. 

44 

54. 

97 

1. 

68 

GO 

59. 

99 

1. 

05 

59. 

99 

1. 

31 

59. 

98 

1. 

57 

59. 

97 

1. 

83 

G5 

64. 

99 

1. 

13 

64. 

98 

1. 

42 

64. 

98 

1. 

70 

64. 

97 

1. 

99 

70 

69. 

99 

1. 

22 

69. 

98 

1. 

53 

69. 

98 

1. 

83 

69. 

97 

2. 

14 

75 

74. 

99 

1. 

31 

74. 

98 

1. 

64 

74. 

97 

1. 

96 

74. 

97 

2. 

29 

80 

79. 

99 

1. 

40 

79. 

98 

1. 

75 

79. 

97 

2, 

09 

79. 

96 

2. 

44 

85 

84. 

99 

1. 

48 

84. 

98 

1. 

85 

84. 

97 

2. 

23 

84. 

96 

2. 

60 

90 

89. 

99 

1. 

57 

89. 

98 

1. 

96 

89. 

97 

2. 

36 

89. 

96 

2. 

75 

'95 

94. 

99 

1. 

66 

94. 

98 

2. 

07 

94. 

97 

2. 

49 

94. 

96 

2. 

90 

100 

99. 

98 

1. 

75 

99. 

98 

2. 

18 

99. 

97 

2. 

62 

99. 

95 

3. 

05 

Distance. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

89 Deg. 

83% Deg. 

88% Deg. 

88% Beg. 












































































TRAVERSE TABLE. 73 


£ 

a 
a 
•*— * 

2 Deg. 

2% Deg. 

2% Deg. 

2% Deg. 

eo 

S 

Lat. 

Dep. 

Lat.*- 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1. 

00 

0 . 

03 

1 . 

00 

0 . 

04 

1 . 

00 

0 . 

04 

1 . 

00 

0 . 

05 

2 

2. 

00 

0 . 

07 

2. 

00 

0 . 

03 

2, 

00 

0 . 

09 

2. 

00 

0 . 

10 

3 

3. 

00 

0 . 

10 

3. 

00 

0 . 

12 

3. 

00 

0 . 

13 

3. 

00 

0 . 

14 

4 

4. 

00 

0 . 

14 

4. 

00 

0 . 

16 

4. 

00 

0 . 

17 

4. 

00 

0 . 

19 i 

5 

5. 

00 

0 . 

17 

5. 

00 

0 . 

20 

5. 

00 

0 . 

22 

4. 

99 

0 . 

24 j 

6 

6. 

00 

0 . 

21 

6. 

00 

0 . 

24 

5. 

99 

0 . 

26 

5. 

99 

0 . 

29 

7 

7. 

00 

0 . 

24 

G. 

99 

0 . 

27 

6. 

99 

0 . 

31 

6. 

99 

0 . 

34 

8 

7. 

99 

0 . 

28 

7. 

99 

0 . 

31 

7. 

99 

0 . 

35 

7. 

99 

0 . 

38 

9 

8. 

99 

0 . 

31 

e. 

99 

0 . 

35 

8. 

99 

0 . 

39 

8. 

99 

0 . 

43 

10 

9. 

99 

0 . 

35 

9. 

99 

0 . 

39 

9. 

99 

0 . 

44 

9. 

99 

0 . 

48 

11 

10. 

99 

0 . 

38 

10. 

99 

0 . 

43 

10. 

99 

0 . 

48 

10. 

99 

0 . 

53 

12 

11. 

99 

0 . 

42 

11. 

99 

0 . 

47 

11. 

99 

0 . 

52 

11. 

99 

0 . 

53 

13 

12. 

99 

0 . 

45 

12. 

99 

0 . 

51 

12. 

99 

0 . 

57 

12. 

99 

0 . 

62 

14 

13. 

99 

0 . 

49 

13. 

99 

0 . 

55 

13. 

99 

0 . 

61 

13. 

98 

0 . 

67 

15 

14. 

99 

0 . 

52 

14. 

99 

0 . 

59 

14. 

99 

0 . 

65 

14 • 

98 

0 . 

72 

1G 

15. 

99 

0 . 

56 

15. 

99 

0 . 

63 

15. 

99 

0 . 

70 

15. 

98 

0 . 

77 

17 

16. 

99 

0 . 

59 

16. 

99 

0 . 

67 

16. 

98 

0 . 

74 

16. 

98 

0 . 

82 

18 

17. 

99 

0 . 

63 

17. 

99 

0 . 

71 

17. 

98 

0 . 

79 

17. 

98 

0 . 

86 

19 

18. 

99 

0 . 

66 

18. 

99 

0 . 

75 

18. 

98 

0 . 

83 

18. 

98 

0 . 

91 

20 

19. 

99 

0 . 

70 

19. 

98 

0 . 

79 

19. 

98 

0 

87 

19. 

98 

0 . 

96 

21 

20. 

99 

0 . 

73 

20. 

98 

0 . 

82 

20. 

98 

0 . 

92 

1 20. 

98 

1 . 

01 

22 

21. 

99 

0 . 

77 

21. 

93 

0 . 

86 

21. 

98 

0 . 

96 

21. 

97 

1 . 

06 

33 

22. 

99 

0 . 

80 

22. 

98 

0 . 

90 

22. 

98 

1 . 

00 

i 22. 

97 

1. 

10 

24 

23. 

99 

0 . 

84 

23. 

98 

0 . 

91 

23. 

98 

1 . 

05 

23. 

97 

1 . 

15 

25 

24. 

98 

0 . 

87 

24. 

98 

0 . 

98 

24. 

98 

1 . 

09 

24- 

97 

1 . 

20 

26 

25. 

98 

0 . 

91 

25. 

98 

1 . 

03 

25. 

98 

1 . 

13 

25. 

97 

1 . 

25 

27 

26. 

98 

0 . 

94 

26. 

98 

1 . 

06 

26. 

97 

1 . 

18 

j 26. 

97 

1 . 

30 

28 

27. 

98 

0 . 

98 

27. 

93 

1 . 

10 

27. 

97 

1 . 

22 

27. 

97 

1 . 

34 

29 

28. 

98 

1 . 

01 

28. 

93 

1 . 

14 

28. 

97 

1 . 

26 

( 28. 

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<£ 

CJ 

Dep. 

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Dep. 

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Dep. 

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c3 

a 

5 

83 Deg. 

87% Deg. 

87% Deg 


87% Beg 

• 


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74 TRAVERSE TABLE. 


Distance. 

3 Deo. 

3% Deg. 

3% Deg. 

t 

'% Deg. j 

Lat. 

Dep. 

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Dep. 

Lat. 

Dep 

' 

Lat. 

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5 

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99 

0 . 

28 

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23 

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88 

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45 

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82 

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56 

90 

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88 

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71 

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86 

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10 

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83 

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49 

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81 

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89 

95 

94. 

87 

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97 

94. 

85 

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39 

94. 

82 

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80 

94. 

80 

6. 

21 

100 

99. 

86 

5. 

23 

99. 

84 

5. 

67 

99. 

81 

6. 

10 

99. 

79 

6. 

54 

| Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

87 Deo. 

86% Deo. 

86% Deo. 

86% Deg 















































































Distance, ooooooo^^oootoi^^coco MtOMMMtctstotstSHihii-iKKKKHKt-i Distance, 

oc^ooiooiooiooiooiocio tooo-QOson^coiaHioca^ooi^ojM^oooooocn^MMHi 


TRAVERSE TABLE. 


75 


4 Deg. 

4% Deg. 

4% Deg. 

i 

4% Deg. 


Lat. 

Dep. 

Lat. 

Dep. 

"V._ - 

La 

t. 

Dep. 

Lat. 

Dep. 

1. 

00 

0. 

07 

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00 

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2. 

00 

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14 

1. 

99 

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17 

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99 

0. 

21 

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99 

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22 

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99 

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24 

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99 

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21 

3. 

99 

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28 

3. 

99 

0. 

30 

3. 

99 

0. 

31 

i 3. 

98 

t 

33 

4. 

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35 

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37 

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5. 

98 

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63 

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63 

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73 

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82 

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94 

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70 

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93 

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80 

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90 

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94 

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67 

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78 

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93 

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88 

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81 

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93 

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88 

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93 

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95 

27. 

92 

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08 

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91 

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20 

| 27. 

90 

2. 

32 

28. 

93 

2. 

02 

28. 

92 

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15 

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91 

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28 

28. 

90 

2. 

40 

29. 

93 

2. 

09 

29. 

92 

2. 

22 

29. 

91 

2. 

35 

29. 

90 

2. 

48 

34. 

91 

2. 

44 

34. 

90 

2. 

59 

34. 

89 

2. 

75 

34. 

88 

2. 

90 

39. 

90 

2. 

79 

39. 

89 

2. 

96 

39. 

88 

3. 

14 

39. 

86 

3. 

31 

44. 

89 

3. 

14 

44. 

88 

3. 

33 

44. 

86 

3. 

53 

44. 

85 

3. 

73 

49. 

88 

3. 

49 

49. 

86 

3. 

71 

49. 

85 

3. 

92 

49. 

83 

4. 

14 

54. 

87 

3. 

84 

54. 

85 

4. 

08 

54. 

83 

4. 

32 

54. 

81 

4. 

55 

59. 

85 

4. 

19 

59. 

84 

4. 

45 

59. 

82 

4. 

71 

59. 

79 

4. 

97 

64. 

84 

4. 

53 

64. 

82 

4. 

82 

64. 

80 

5. 

10 

64. 

78 

5. 

38 

69. 

83 

4. 

83 

69. 

81 

5. 

19 

69. 

78 

5. 

49 

69. 

76 

5. 

80 

74. 

82 

5. 

23 

74. 

79 

5. 

56 

74. 

77 

5. 

88 

74. 

74 

6. 

21 

79. 

81 

5. 

58 

79. 

78 

5. 

93 

79. 

75 

6. 

28 

79. 

73 

6. 

62 

84. 

79 

5. 

93 

84. 

77 

6. 

30 

84. 

74 

6. 

67 

84. 

71 

7. 

04 

89. 

78 

6. 

28 

89. 

75 

6. 

67 

89. 

72 

7. 

06 

89. 

69 

7. 

45 

94. 

77 

6. 

63 

94. 

74 

7. 

04 

94. 

71 

7. 

45 

94. 

67 

7. 

87 

99. 

76 

6. 

98 

99. 

73 

7. 

41 

99. 

69 

7. 

85 

99. 

66 

8. 

28 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

86 Deg. 

85% Deg 


85% Deg 

• 

GO 

Deg 

* 



































































TRAVERSE TABLE. 


QJ 

V 

C 

a 

5 Deg. 


f>34 Deg 


544 Deg. 

F 

t 

% Deg. 


co 

c 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

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1 

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4 

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98 

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38 

3. 

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0. 

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5 

4. 

98 

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44 

4. 

98 

0. 

46 

4. 

98 

0. 

48 

4. 

97 

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50 

6 

5. 

98 

0. 

52 

5. 

97 

0. 

55 

5. 

97 

0. 

58 

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60 

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6. 

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73 

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96 

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76 

7. 

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14 

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37 

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93 

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63 

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91 

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70 

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92 

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73 

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91 

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80 

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66 

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74 

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82 

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£0 

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10 

22 

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92 

1. 

92 

21. 

91 

2. 

01 

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90 

2. 

11 

21. 

89 

2. 

20 

23 

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91 

2. 

00 

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90 

2. 

10 

22. 

89 

2. 

20 

22. 

88 

2. 

30 

24 

23. 

91 

2. 

09 

23. 

90 

2. 

20 

23. 

89 

2. 

30 

23. 

88 

2. 

40 

25 

24. 

90 

2. 

18 

24. 

90 

2. 

29 

24. 

88 

2. 

40 

24. 

87 

2. 

50 

26 

25. 

90 

2. 

27 

25. 

89 

2. 

38 

25. 

88 

2. 

49 

25. 

87 

2. 

60 

27 

26. 

90 

2. 

35 

26. 

89 

2. 

47 

26. 

88 

2. 

59 

26. 

86 

2. 

71 

28 

27. 

89 

2. 

44 

27. 

88 

2. 

56 

27. 

87 

2. 

68 

27. 

86 

2. 

81 

29 

28. 

89 

2. 

53 

28. 

83 

2. 

65 

28. 

87 

2. 

78 

28. 

85 

2. 

91 

30 

29. 

89 

2- 

61 

29. 

87 

2. 

75 

29. 

86 

2. 

88 

29. 

85 

3. 

01 

35 

31. 

87 

3. 

05 

34. 

85 

3. 

20 

34. 

84 

3. 

35 

34. 

82 

3. 

51 

40 

39. 

85 

3- 

49 

39. 

83 

3. 

66 

39. 

82 

3. 

83 

39. 

80 

4. 

01 

45 

44. 

83 

3. 

92 

44. 

81 

4. 

12 

44. 

79 

4. 

31 

44. 

77 

4. 

51 

50 

49. 

81 

4. 

36 

49. 

79 

4. 

58 

49. 

77 

4. 

79 

49. 

75 

5. 

01 

55 

54. 

79 

4- 

79 

54. 

77 

5. 

03 

54. 

75 

5. 

27 

54. 

72 

5. 

51 

GO 

59. 

77 

5. 

23 

59. 

75 

5. 

49 

59. 

72 

5. 

75 

59. 

70 

6. 

01 

65 

64. 

75 

5. 

67 

64. 

73 

5. 

95 

64. 

70 

6. 

23 

64. 

67 

6. 

51 

70 

69. 

73 

6. 

10 

69. 

71 

0. 

41 

69. 

68 

6. 

71 

69. 

65 

7. 

01 

75 

74. 

71 

6. 

54 

74. 

69 

6. 

06 

74. 

65 

7. 

19 

74. 

62 

7. 

51 

80 

79. 

70 

o. 

97 

79. 

66 

7. 

32 

79. 

63 

7. 

67 

79. 

60 

8. 

02 

85 

84. 

68 

7- 

41 

84. 

64 

7. 

78 

84. 

61 

8. 

15 

84. 

57 

8. 

52 

90 

89. 

66 

7- 

84 

89. 

62 

8. 

24 

89. 

59 

8. 

63 

89. 

55 

9. 

02 

95 

94. 

64 

8. 

28 

04. 

60 

8. 

69 

94. 

56 

9. 

11 

94. 

52 

9. 

52 

100 

99. 

62 

8. 

72 

99. 

53 

9. 

15 

99. 

54 

9. 

58 

99. 

50 

10. 

02 

6 

o 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

q 

85 Deg. 

N* 

©0\ 

GO 

Deg 

. 

84^4 Deg 


84)4 Deg 




































































TRAVERSE TABLE. 77 


O' 

o 

c 

Cj 

•*-> 

6 Deg. 



Deg. 


6 ^ Deg. 

6 % Deg. 

!C 

5 

Lat. 

Dep. 

Lat. 

Dep. 

i. 

r* 

Dep. 

Lat. 

Dep. 

1 

0 . 

99 

0 . 

10 

0 . 

99 

0 . 

11 

0 . 

99 

0 . 

11 

0 . 

99 

0 . 

12 

2 

1 . 

99 

0 . 

21 

1 . 

99 

0 . 

22 

1 . 

99 

0 . 

23 

1 . 

99 

0 . 

24 

3 

2 . 

98 

0 . 

31 

2 . 

98 

0 . 

33 

2 . 

98 

0 . 

34 

2 . 

98 

0 . 

35 

4 

3. 

98 

0 . 

41 

3. 

98 

0 . 

44 

3. 

97 

0 . 

45 

3. 

97 

0 . 

47 

5 

4. 

97 

0 . 

52 

4. 

97 

0 . 

54 

4. 

97 

0 . 

57 

4. 

97 

0 . 

59 

0 

5. 

97 

0 . 

63 

5. 

96 

0 . 

65 

5. 

96 

0 . 

68 

5. 

96 

0 . 

71 | 

7 

6 . 

96 

0 . 

73 j 

6 . 

96 

0 . 

76 

6 . 

96 

0 . 

79 

6 . 

95 

0 . 

82 

8 

7. 

96 

0 . 

84 

7. 

95 

0 . 

87 

7. 

95 

0 . 

91 

7. 

94 

0 . 

94 

9 

8 . 

95 

0 . 

94 

8 . 

95 

0 . 

93 

8 . 

94 

1 . 

02 

8 . 

94 

1 . 

06 

10 

9. 

95 

1 . 

05 j 

9 

91 

1 . 

09 

9. 

94 

1 . 

13 

9. 

93 

1 . 

18 

11 

10 . 

94 

1 . 

15 

10 . 

93 

1 . 

20 

10 . 

93 

1 . 

25 

10 . 

92 

1 . 

29 

12 

11 . 

93 

1 . 

25 | 

11 . 

93 

1 . 

31 

11 . 

92 

1 . 

36 

11 . 

92 

1 . 

41 

13 

12 . 

93 

1 . 

36 

12 . 

92 

1 . 

42 

12 . 

92 

1 . 

47 

12 . 

91 

1 . 

53 

14 

13. 

92 

1 . 

46 

13. 

92 

1 . 

52 

13. 

91 

1 . 

59 

13. 

90 

1 . 

65 

15 

14. 

92 

1 . 

57 

14. 

91 

1 . 

63 

11 . 

90 

1 . 

70 

14. 

90 

1 . 

76 

16 

15. 

91 

1 . 

67 

15. 

90 

1 . 

74 

15. 

90 

1 . 

81 

15. 

89 

1 . 

83 

17 

16. 

91 

1 . 

78 

16. 

90 

1 . 

85 

13. 

89 

1 . 

92 

16. 

88 

2 . 

00 

18 

17. 

90 

1 . 

88 

17. 

89 

1 . 

96 

17. 

88 

2 . 

04 

17- 

88 

2 . 

12 

19 

18. 

90 

1 . 

99 

18. 

89 

2 . 

07 

13. 

88 

2 . 

15 

18. 

87 

2 . 

23 

20 

19. 

89 

2 . 

09 

19. 

88 

2 . 

18 

19. 

87 

2 . 

26 

19. 

86 

2 . 

35 

21 

20 . 

83 

2 . 

20 

20 . 

88 

2 . 

29 

20 . 

87 

2 . 

33 

20 . 

85 

2 . 

47 

22 

21 . 

83 

2 . 

30 

21 . 

87 

2 . 

40 

21 . 

86 

2 . 

49 

21 . 

85 

2 . 

59 

23 

22 . 

87 

2 . 

40 

22 . 

86 

2 . 

50 

22 . 

85 

2 . 

60 

22 . 

84 

2 . 

70 

24 

23. 

87 

2 . 

51 

23. 

86 

2 . 

61 

23. 

85 

2 . 

72 

23. 

83 

2 . 

82 

25 

24. 

86 

2 . 

61 

24. 

83 

2 . 

72 

24. 

81 

2 . 

83 

24. 

83 

2 . 

Cl 

26 

25. 

86 

2 . 

72 

25. 

85 

2 . 

83 

25. 

83 

2 . 

94 

25. 

82 

3. 

03 

27 

26. 

85 

2 . 

82 

26. 

84 

2 . 

94 

26. 

83 

3. 

06 

26. 

81 

3. 

17: 

23 

27. 

85 

2 . 

93 

27. 

83 

o 

t). 

05 

27. 

82 

3. 

17 

27. 

81 

3. 

20 . 

29 

28. 

84 

3. 

03 

28. 

83 

3. 

16 

28. 

81 

3. 

28 

28. 

80 

3. 

41 

30 

29. 

84 

3. 

14 

! 29. 

82 

3. 

27 

29. 

61 

3. 

40 

29. 

79 

3. 

53 

35 

31. 

81 

3. 

66 

34. 

79 

3. 

81 

34. 

73 

3. 

96 

34. 

76 

4. 

11 i 

40 

30. 

73 

4. 

13 

39. 

76 

4. 

35 

30. 

74 

4. 

53 

89. 

72 

4. 

70 

45 

41. 

75 

4. 

70 

44. 

73 

4. 

90 

44. 

71 

5. 

09 

44. 

69 

5. 

29 ! 

50 

43. 

73 

5. 

23 

49. 

70 

5. 

44 

49. 

63 

5. 

63 

49. 

65 

5. 

83 

55 

54. 

70 

5. 

75 

54. 

67 

5. 

99 

54. 

65 

6 . 

23 

j 54. 

62 

6 . 

46; 

60 

59. 

67 

6 . 

27 

59. 

64 

6 . 

53 

59. 

61 

6 . 

79 

i 59. 

58 

7. 

05 ! 

65 

61. 

61 

6 . 

79 

64. 

61 

7. 

08 

61. 

53 

7. 

33 

| 64. 

55 

7. 

64 

70 

63. 

62 

7. 

33 

69. 

53 

7. 

62 

69. 

55 

7. 

92 

69. 

51 

8 . 

23 

75 

74. 

53' 

7. 

84- 

74. 

55 

8 . 

17 

74. 

52 

8 . 

49 

74. 

48 

8 . 

82 

80 

79. 

56 

8 . 

33 

79. 

53 

8 . 

71 

79. 

49 

9. 

06 

79. 

45 

9. 

40 

85 

84. 

53 

8 . 

83 

84. 

50 

9. 

25 

84. 

45 

9. 

62 

84. 

41 

9. 

99 

90 

89. 

51 

9. 

41 

89. 

47 

9. 

80 

89. 

42 

10 . 

19 

89. 

38 

10 . 

58 

95 

94. 

43 

9. 

93 

94. 

41 

10 . 

84 

94. 

39 

10 . 

75 

91. 

34 

11 . 

17 

100 

99. 

45 

10 . 

45 

99. 

41 

10 . 

80 

99. 

36 

11 . 

32 

99. 

81 

11 . 

75 

<V 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

c3 

-*-> 

Xj 

5 

84 Deg. 

8 


Deg. 


83 X 

Deg. 


CO 

co 

Deg 

• 



























































































78 TRAVERSE TABLE. 


<6 

o 

c 

C3 

7 Deo. 

7 )i Deg. 

7)4 Deg. 

7% Deg. 

(/} 

A 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 

99 

0. 

12 

0. 

99 

0. 

13 

0. 

99 

0. 

13 

0. 

99 

0. 

13 

2 

1. 

99 

0. 

24 

1. 

98 

0. 

23 

1. 

93 

0. 

26 

1. 

98 

0. 

27 

3 

2 . 

98 

0. 

37 

2 . 

98 

0. 

33 

2 . 

97 

0. 

39 

2 . 

97 

0. 

40 

4 

3. 

97 

0. 

49 

3. 

97 

0. 

50 

3. 

97 

0. 

52 

3. 

96 

0. 

54 

5 

4. 

£6 

0. 

61 

4. 

96 

0. 

63 

4. 

96 

0. 

65 

4. 

95 

0. 

67 

6 

5. 

96 

0. 

73 

5. 

95 

0. 

76 

5. 

95 

0. 

*8 

5. 

OX 

c' U 

0. 

81 

7 

6 . 

95 

0. 

85 

6 . 

94 

0. 

88 

6 . 

94 

0 . 

91 

6 . 

94 

0. 

94 

8 

7. 

94 

0. 

97 

7. 

94 

1. 

01 

7. 

93 

1. 

04 

7. 

93 

1. 

08 

9 

8 . 

93 

1. 

10 

8 . 

93 

1. 

14 

8 . 

92 

1. 

17 

8 . 

CO 

1. 

21 

10 

9. 

93 

1. 

22 

9. 

92 

1. 

26 

9. 

91 

1. 

81 

9. 

91 

1. 

35 

11 

10 . 

92 

1. 

34 

10 . 

91 

1. 

39 

10 . 

91 

1. 

44 

10 . 

90 

1. 

48 

12 

11 . 

91 

1. 

40 

11 . 

90 

1. 

51 

11 . 

90 

1. 

57 

11 . 

89 

1. 

62 

13 

12 . 

90 

1. 

53 

12 . 

90 

1. 

64 

12 . 

89 

1. 

70 

12 . 

88 

1. 

75 

14 

13. 

90 

1. 

71 

13. 

89 

1. 

77 

13. 

88 

1. 

83 

13. 

87 

1. 

89 

15 

14. 

89 

1. 

83 

14. 

88 

1. 

89 

14. 

87 

1. 

96 

14. 

86 

2 . 

02 

16 

15. 

88 

1. 

95 

15. 

87 

2 . 

02 

15. 

86 

2 . 

09 

15. 

85 

2 . 

16 

17 

16. 

87 

2 . 

07 

16. 

86 

2 . 

15 

16. 

85 

2 . 

22 

16. 

84 

2 . 

29 

18 

17. 

87 

2 . 

19 

17. 

86 

2 . 

27 

17. 

85 

2 . 

35 

17. 

84 

2 . 

43 

19 

18. 

86 

2 . 

32 

18. 

85 

2 . 

40 

13. 

84 

2 . 

48 

18. 

83 

2 . 

56 

20 

19. 

85 

2 . 

44 

19. 

84 

2 . 

52 

19. 

83 

2 . 

61 

19. 

82 

2 . 

70 

21 

20 . 

84 

2 . 

56 

20 . 

83 

2 . 

65 

20 . 

82 

2 . 

74 

20 . 

81 

2 . 

83 

22 

21 . 

84 

2 . 

68 

21 . 

82 

2 . 

78 

21 . 

81 

2 . 

87 

21 . 

80 

2 . 

97 

23 

22 . 

83 

2 . 

80 

22 . 

82 

2 . 

90 ! 

22 . 

80 

3. 

00 

22 . 

79 

3. 

10 

24 

23. 

82 

2 . 

92 

23. 

81 

3. 

03 

23. 

79 

3. 

13 

| 23. 

78 

3. 

24 

25 

24. 

81 

3. 

05 

24. 

80 

3. 

15 

24. 

79 

3. 

26 

24. 

77 

3. 

37 

26 

25. 

81 

3. 

17 

25. 

79 

3. 

28 

25. 

78 

3. 

39 

25. 

76 

3. 

51 

27 

26. 

89 

3. 

29 

26. 

78 

3. 

41 

26. 

77 

3. 

52 

26. 

75 

3. 

64 

28 

27. 

79 

3. 

41 

27. 

78 

3. 

53 

27. 

76 

3. 

65 

27. 

74 

3. 

78 

29 

28. 

78 

3- 

53 

28. 

77 

3. 

66 

28. 

75 

3. 

79 

! 28. 

74 

3. 

91 

30 

29. 

78 

3. 

66 

29. 

76 

3. 

79 

29. 

74 

3. 

92 

: 29. 

73 

4. 

05 

35 

34. 

74 

4. 

27 

34. 

72 

4. 

42 

34. 

70 

4. 

57 

34. 

68 

4. 

72 

40 

39. 

70 

4. 

87 

39. 

68 

5. 

05 

39. 

66 

5. 

22 

39. 

63 

5. 

39 

45 

44. 

67 

5- 

48 

44. 

64 

5. 

63 

44. 

62 

5. 

87 

44. 

59 

6 . 

07 

50 

49. 

63 

6 . 

09 

49. 

60 

6 . 

31 

49. 

57 

6 . 

53 

49. 

54 

6 . 

74 

55 

54. 

59 

6 . 

70 

54. 

56 

6 . 

94 

54. 

53 

7. 

18 

54. 

50 

7. 

42 

60 

59. 

55 

7- 

31 

59. 

52 

7. 

57 

59. 

49 

7. 

83 

59. 

45 

8 . 

09 

65 

64. 

52 

7. 

92 

64. 

48 

8 . 

20 

64. 

44 

8 . 

48 

64. 

41 

8 . 

77 

70 

69. 

48 

8 . 

53 

69. 

44 

8 . 

83 

69. 

40 

9. 

14 

69. 

36 

9. 

44 

75 

74. 

44 

9- 

14 

74. 

40 

9. 

46 

74. 

36 

9. 

79 

74. 

31 

10 . 

11 

80 

79. 

40 

9. 

75 

79. 

36 

10 . 

10 

79. 

32 

10 . 

44 

79. 

27 

10 . 

79 

85 

84. 

37 

10 . 

36 

84. 

32 

10 . 

73 

84. 

27 

11 . 

09 

84. 

22 

11 . 

46 

90 

89, 

33 

10 - 

97 

89. 

28 

11 . 

36 

89. 

23 

11 . 

75 

89. 

18 

12 . 

14 

95 

94. 

29 

11 . 

58 

94. 

24 

11 . 

99 

94. 

19 

12 . 

40 

94. 

13 

12 . 

81 

100 

99. 

25 

12 . 

19 

99. 

20 

12 . 

62 

99. 

14 

13. 

05 

99. 

09 

13. 

49 

© 

O 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

l 

cl 

5 

83 Deg. 


82 % 

Deg 

• 

82)4 

Deg 

• 

82)^ Deg 


























































































TRAVERSE 

TABLE. 


79 

*• 

o 

a 

c3 

■*-> 

8 Deo. 

8% Deg. 

8 % Deg. 

8% Deg. 

Ji 

3 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| 

Dep. 

1 

0. 99 

0. 14 

0. 99 

0. 14 

0. 99 

0. 15 

0. 99 

0. 15 

2 

1. 98 

0. 23 

1. 98 

0. 29 

1. 98 

0. 30 

1. 98 

0. 30 

3 

2. 97 

0. 42 

2. 97 

0. 43 

2. 97 

0. 44 

2. 97 

0. 46 

4 

3. 96 

0. 56 

3. 96 

0. 57 

3. 96 

0. 59 

3. 95 

0. 61 

5 

4. 95 

0. 70 

4. 95 

0. 72 

4. 95 

0. 74 

4. 94 

0. 76 

6 

5. 94 

0. 84 

5. 94 

0. 86 

5. 93 

0. 89 

5. 93 

0. 91 

7 

6. 93 

0. 97 

6. 93 

1. 00 

6. 92 

1. 03 

6. 92 

1. 06 

8 

7. 92 

1. 11 

7. 92 

1. 15 

7. 91 

1. 18 

7. 91 

1. 22 

9 

8. 91 

1. 25 

8. 91 

1. 29 

8. 90 

1. 33 

8. 90 

1. 37 

10 

9. 90 

1. 30 

9. 90 

1 43 

9. 89 

1. 48 

9. 88 

1. 52 

11 

10. 89 

1. 53 

10. 89 

1. 53 

10. 88 

1. 63 

10. 87 

1. 67 

12 

11. 88 

1. 67 

11. 88 

1. 72 

11. 87 

1. 77 

11. 86 

1. 83 

13 

12. 87 

1. 81 

12. 87 

1. 87 

12. 86 

1. 92 

12. 85 

1. 98 

14 

13. 86 

1. 95 

13. 86 

2. 01 

13. 85 

2. 07 

13. 84 

2. 13 

15 

14. 85 

2. 09 

14. 85 

2. 15 

14. 84 

2. 22 

14. 83 

2. 28 

16 

15. 84 

2. 23 

15. 84 

2. 30 

15. 82 

2. 36 

15. 81 

2. 43 

17 

16. 83 

2. 37 

16. 83 

2. 44 

16. 81 

2. 51 

16. 80 

2. 59 

18 

17. 82 

2. 51 

17. 81 

2. 58 

17. 80 

2. 66 

17. 79 

2. 74 

19 

18. 82 

2. 64 

18. 80 

2. 73 

18. 79 

2. 81 

18. 78 

2. 89 

20 

19. 81 

2. 78 

19. Y9 

2. 87 

19. 78 

2. 96 

19. 77 

3. 04 

21 

20. 80 

2. 92 

20. 78 

3. 01 

20. 77 

3. 10 

20. 76 

3. 19 

22 

21. 79 

8. 06 

21. 77 

3. 13 

21. 76 

3. 25 

21. 74 

3. 35 

23 

22. 73 

3. 20 

22. 76 

3. 30 

22. 75 

3. 40 

22. 73 

3. 50 

24 

23. 77 

3. 34 

23. 75 

3. 44 

23. 74 

3. 55 

! 23. 72 

3. 65 

25 

24. 76 

3. 48 

24. 74 

3. 59 

24. 73 

3. 70 

! 24. 71 

3. 80 

26 

25. 75 

3. 62 

25. 73 

3. 73 

25. 71 

3. 84 

25. 70 

3. 96 

27 

26. 74 

3. 76 

26. 72 

3. 87 

26. 70 

3. 99 

26. 69 

4. 11 

23 

27. 73 

3. 90 

27. 71 

4. 02 

27. 69 

4. 14 

27. 67 

4. 26 

29 

28. 72 

4. 04 

28. 70 

4. 16 

28. 68 

4. 29 

28. 66 

4. 41 

30 

29. 71 

4. 18 

29. 69 

4. 30 

29. 67 

4. 43 

29. 65 

4. 56 

35 

34. 66 

4. 87 

34. 64 

5. 02 

34. 62 

5. 17 

34. 59 

5. 32 

40 

39. 61 

5. 57 

39. 59 

5. 74 

39. 56 

5. 91 

39. 53 

6. 08 

45 

44. 56 

6. 23 

44. 53 

6. 46 

44. 51 

6. 65 

44. 48 

6. 85 

50 

49. 51 

6. 96 

49. 48 

7. 17 

49. 45 

7. 39 

49. 42 

7. 61 

55 

54. 46 

7. 65 

54. 43 

7. 89 

54. 40 

8. 13 

54. 36 

8. 37 

60 

59. 42 

8. 35 

59. 38 

8. 61 

59. 34 

8. 87 

59. 30 

9. 13 

65 

64. 37 

9. 05 

64. 33 

9. 33 

64. 29 

9. 61 

64. 24 

9. 89 

70 

69. 32 

9. 74 

69. 28 

10. 04 

69. 23 

10. 35 

69. 19 

10. 65 

75 

74. 27 

10. 44 

74. 22 

10. 76 

74. 18 

11. 09 

74. 13 

11.41 

80 

79. 22 

11. 13 

79. 17 

11. 48 

79. 12 

11. 82 

79. 07 

12. 17 

85 

84. 17 

11. 83 

84. 12 

12. 20 

84. 07 

12. 56 

84. 01 

12. 93 

90 

89. 12 

12. 53 

89. 07 

12. 91 

89. 01 

13. 30 

88. 95 

13. 69 

95 

94. 08 

13. 22 

94. 02 

13. 63 

93. 96 

14. 04 

93. 89 

14. 45 

100 

99. 03 

13. 92 

98. 97 

14. 35 

98. 90 

14. 78 

98. 84 

15. 21 

6 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

a 

00 

5 

•82 Deo. 

81% Deg. 

81% Deg. 

813^ Deg. 


i 






































































80 




TRAVERSE 

TABLE. 



-- 

Distance. 

9 Deg. 

9% Deg. 


9% Deg. 

P- 

9% Deg. 

Lat. 

Dep. 

Lat. 

.... 

Dep. 

Lat. 

D< 

Lat. 

Dep. 

1 

0. 99 

0. 

16 

0 . 

99 

0. 16 

0 . 

99 

0 . 

17 

0. 99 

0. 17 I 

2 

1. 98 

0 . 

81 

1. 

97 

0. 32 

1. 

97 

0 . 

33 

1. 97 

0. g4 !j 

3 

2. 96 

0 . 

47 

2. 

96 

0. 43 

2. 

96 

0 . 

50 

2. 96 

0. 51 1 

4 

3. 95 

0 . 

63 

3. 

95 

0. 64 

3. 

95 

0 . 

66 

3. 94 

0. 68 i 

5 

4. 94 

0 . 

78 

4. 

93 

0. 80 

4. 

93 

0 . 

83 

4. 93 

0. 85 

G 

5. S3 

0 . 

94 

5. 

92 

0. 96 

5. 

92 

0 . 

99 

5. 91 

1. 02 

7 

6. 91 

1. 

10 

6. 

91 

1. 13 

6. 

90 

1. 

16 

6. 90 

1. 19 

8 

7. 90 

1 . 

25 

7. 

90 

1. 29 

7. 

89 

1. 

32 

7. 88 

1. 35 

9 

8. 80 

1. 

41 

8. 

83 

1. 45 

8. 

88 

1. 

49 

8. 87 

1. 52 

10 

9. 88 

1. 

56 

9. 

87 

1. 61 

9. 

86 

1. 

65 

9. 86 

1. 69 

11 

10. 8G 

1. 

72 

10. 

86 

1. 77 

10. 

85 

1. 

82 

10. 84 

1. 86 

12 

11. 85 

1. 

88 

11. 

84 

1. 93 

n. 

84 

1. 

98 

11. 83 

2. 03 

13 

12. 84 

2. 

03 

12. 

83 

2. 09 

12. 

82 

2. 

15 

12. 81 

2. 20 

14 

13. 83 

2. 

19 

13. 

82 

2. 25 

13. 

81 

2. 

31 

13. 80 

2. 37 

15 

14. 82 

2. 

35 

14. 

80 

2. 41 

14. 

70 

2. 

48 

14. 78 

2. 54 

1G 

15. 80 

2. 

50 

15. 

79 

2. 57 

15. 

73 

2. 

64 

15. 77 

2. 71 

17 

16. 79 

2. 

66 

16. 

73 

2. 73 

16. 

77 

2. 

81 

16. 75 

2. 88 

18 

17. 73 

2. 

82 

17. 

77 

2. 89 

17. 

75 

2. 

97 

17. 74 

3. 05 

19 

18. 77 

2. 

97 

18. 

75 

3. 05 

18. 

74 

3. 

14 

18. 73 

3. 22 

20 

19. 75 

3. 

13 

19. 

74 

3. 21 19. 

73 

3. 

30 

19. 71 

3. 39 

21 

20. 74 

3. 

29 

20. 

73 

3. 33 | 

20. 

71 

3. 

47 

20. 70 

3. 56 

22 

21..73 

3. 

44 

21. 

71 

3. 54 

21. 

70' 

3. 

63 

21. 68 

3. 73 

23 

22. 72 

3. 

60 

22. 

70 

3. 70 

22. 

68 

3. 

80 

22. 67 

3. 90 

24 

23. 70 

3. 

75 

23. 

69 

3. 86 

23. 

67 

3. 

96 

23. 65 

4. 06 

25 

24. 69 

3. 

91 

24. 

67 

4. 02 

24. 

66 

4. 

13 

24. 64 

4. 23 

26 

25. 63 

4. 

07 

25. 

66 

4. 18 

25. 

64 

4. 

29 

25. 62 

4. 40 

27 

26. 67 

4. 

22 

26. 

65 

4. 34 

26. 

63 

4. 

46 

26. 61 

4. 57 

28 

27. 66 

4. 

38 

27. 

64 

4. 50 

27. 

62 

4. 

62 

27. 60 

4. 74 

29 

28. 64 

4. 

54 

28. 

62 

4. 66 

28. 

CO 

4. 

79 

28. 58 

4. 91 

30 

29. 63 

4. 

69 

29. 

G1 

4. 82 

29. 

59 

4. 

95 

29. 57 

5. 08 

35 

34. 57 

5. 

43 

34. 

54 

5. 63 i 

34. 

52 

5. 

78 

34. 49 

5. 93 

40 

39. 51 

6. 

26 

39. 

48 

6. 43 

39. 

45 

6. 

60 

39. 42 

6. 77 

45 

44. 45 

7. 

04 

44. 

41 

7. 23 

44. 

33 

7. 

43 

44. 35 

7. 62 

50 

49. 33 

7. 

82 

49. 

35 

8. 04 

49. 

32 

8. 

25 

49. 28 

8. 47 

55 

54. 32 

8. 

60 

54. 

28 

8. 84 

54. 

25 

9. 

08 

54. 21 

9. 31 

GO 

59. 26 

9. 

39 

59. 

22 

9. 64 

59. 

18 

9. 

90 

59. 13 

10. 16 

G5 

64. 20 

10. 

17 

64. 

15 

10. 45 

64. 

11 

10. 

73 

64. 06 

11. 01 

70 

69. 14 

10. 

95 

69. 

09 

11. 25 

69. 

04 

11. 

55 

68. 99 

11. 85 

75 

74. 08 

11. 

73 

74. 

02 

12. 06 

73. 

97 

12. 

38 

73. 92 

12. 70 

80 

79. 02 

12. 

51 

78. 

96 

12. 86 

78. 

90 

13. 

20 

78. 84 

13. 55 

85 

83. 95 

13. 

30 

83. 

89 

13. 66 

83. 

83 

14. 

03 

83. 77 

14. 39 

90 

88. 89 

14. 

08 

88. 

83 

14. 47 

88. 

77 

14. 

85 

88. 70 

15. 24 

95 

93. 83 

14. 

86 

93. 

76 

15. 27 

93. 

70 

15. 

68 

93. 63 

16. 09 

100 

98. 77 

15. 

64 

98. 

70 

16. 07 

98. 

63 

16. 

50 

98. 56 

16. 93 

I Distance. 

1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

81 Deg. 

80% Deg. 

80% Deg. 

£0% Deg. 
















































































Distance. occacc^^fQOCiottKi^osM Distance. 

OOlOOlOOlOOiOOlOOlOOlO ©QO<JOiOn^Mi.'OMOCOOO<iC5Crt^WtCH‘OCOOO«^OW^05tOH' 


TRAVERSE TABLE. 


81 


10 Deg. 

10% Deg. 


iox 

Deg 


10% 

Deg 

• 

Lat. 

Dep. 

Lat. 

Dep 

1.3 

r. 

Dep. 

Lat. 

Dep. 

0 . 

98 

0 . 

17 

0 . 

98 

0 . 

13 

0 . 

98 

0. 

13 

0. 98 

0. 

19 

1 . 

97 

0 . 

35 

1 . 

97 

0 . 

33 

1 . 

97 

0. 

36 

1. 96 

0 . 

37 

2. 

95 

0 . 

52 

2. 

95 

0 . 

53 

2. 

95 

0 . 

55 

2. 95 

0 . 

56 

3. 

94 

0 . 

69 

3. 

94 

0 . 

71 

3. 

93 

0 . 

73 

3. 93 

0 . 

75 

4. 

92 

0 . 

87 

4, 

92 

0 . 

89 

4. 

92 

0 . 

91 

4. 91 

0 . 

93 

5. 

91 

1 . 

04 

5. 

90 

1 . 

07 

5. 

90 

1 . 

09 

5. 89 

1 . 

12 

6. 

89 

1 . 

22 

G. 

89 

1 . 

25 

6. 

88 

1 . 

28 

6. 88 

1 . 

31 

7. 

88 

1 . 

39 

7. 

87 

1. 

42 

7. 

87 

1 . 

46 

7. 86 

1 . 

49 

8. 

86 

1 . 

56 

8. 

86 

1. 

GO 

8. 

85 

1 . 

64 

8. 84 

1 . 

68 

9. 

85 

1 . 

74 

9. 

84 

1. 

73 

9. 

83 

1 . 

82 

9. 82 

1 . 

87 

10. 

83 

1 . 

91 

10. 

8.3 

1. 

96 

10. 

82 

2. 

00 

10. 81 

2. 

05 

11. 

82 

2. 

08 

11. 

81 

2. 

14 

11. 

80 

2. 

19 

11. 79 

2. 

24 

12. 

80 

2. 

26 

12. 

79 

2. 

31 

12 . 

73 

2. 

37 

12. 77 

2. 

42 

13. 

79 

2. 

43 

13. 

73 

2, 

49 

13. 

77 

2. 

55 

13. 75 

2. 

61 

14. 

77 

2. 

60 

14. 

76 

2. 

67 

14. 

75 

2. 

73 

14. 74 

2. 

80 

15. 

76 

2. 

78 

15. 

74 

2. 

85 

15. 

73 

2. 

92 

15. 72 

2. 

98 

16. 

74 

2. 

95 

16. 

73 

3. 

03 

16. 

72 

3. 

10 

16. 70 

3. 

17 

17. 

73 

3. 

13 

17. 

71 

8. 

20 

17. 

70 

3. 

28 

! 17. 68 

3. 

36 

18. 

71 

3. 

30 

I 18. 

70 

3. 

38 

18. 

63 

3. 

46 

18. 67 

3. 

54 

19. 

70 

3. 

47 

19. 

63 

3. 

56 

19. 

67 

3. 

64 

19. 65 

3. 

73 

20. 

68 

3. 

65 

20. 

66 

3. 

74 

20. 

63 

3. 

83 

20. 63 

3. 

92 

21. 

67 

3. 

82 

21. 

65 

3. 

91 

21. 

63 

4. 

01 

21. 61 

4. 

10 

22. 

65 

3. 

99 

22, 

63 

4. 

09 

22. 

61 

4. 

19 

| 22. 60 

4. 

29 

23. 

64 

4. 

17 

23. 

62 

4. 

27 

23. 

60 

4. 

37 

j 23. 58 

4. 

48 

24. 

62 

4. 

34 

24. 

60 

4. 

45 

24. 

53 

4. 

56 

24. 56 

4. 

66 

25. 

61 

4. 

51 

25. 

59 

4. 

63 

25. 

56 

4. 

74 

25. 54 

4. 

85 

26. 

59 

4. 

69 

26. 

57 

4. 

80 

26. 

55 

4. 

92 

26. 53 

5. 

04 

27. 

57 

4. 

86 

27. 

55 

4. 

98 

27. 

53 

5. 

10 

27. 51 

5. 

22 

28. 

56 

5. 

04 

28. 

54 

5. 

16 

28. 

51 

5. 

28 

| 28. 49 

5. 

41 

29. 

54 

5. 

21 

29. 

52 

5. 

34 

29. 

50 

5. 

47 

29. 47 

5. 

60 

34. 

47 

6. 

08 

34. 

44 

6. 

23 

34. 

41 

6. 

33 

34. 39 

6. 

53 

39. 

39 

6. 

95 

39. 

33 

7. 

12 

39. 

33 

7. 

29 

39. 30 

7. 

46 

44. 

32 

7. 

81 

44. 

28 

8. 

01 

44. 

25 

8. 

20 

44. 21 

8. 

39 

49. 

24 

8. 

63 

49. 

20 

8. 

90 

49. 

16 

9. 

11 

, 49. 12 

9. 

33 

54. 

16 

9. 

55 

54. 

12 

9. 

79 

j 54. 

08 

10. 

02 

54. 03 

10. 

26 

59. 

09 

10. 

42 

59. 

04 

10. 

63 

59. 

00 

10. 

93 

58. 95 

11. 

19 

64. 

01 

11. 

29 

63. 

96 

11. 

57 

i 63. 

91 

11. 

85 

63. 86 

12. 

12 

68. 

94 

12. 

16 

68. 

88 

12. 

46 

68. 

83 

12. 

76 

68. 77 

13. 

06 

73. 

86 

13. 

02 

73. 

80 

13. 

35 1 

73. 

74 

13. 

67 

73. 68 

13. 

99 

78. 

78 

13. 

89 

78. 

72 

14. 

24 I 

78. 

66 

14. 

58 

78. 60 

14. 

92 

83. 

71 

14. 

76 

83. 

64 

15. 

13 I 

83. 

58 

15. 

49 

83. 51 

15. 

85 

88. 

63 

15. 

63 

88. 

56 

16. 

01 j 

88. 

49 

16. 

40 

88. 42 

16. 

79 

93. 

56 

16. 

50 

93. 

48 

16. 

90 

93. 

41 

17. 

31 

93. 33 

17. 

72 

98. 

43 

17. 

36 

98. 

40 

17. 

79 

98. 

33 

18. 

22 

98. 25 

18. 

65 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

80 Deg. 

I 

f- 

< 

•9 H 

Deg. 


WA 

Deg 


79^ 

Deg 

• 







































































o<> 

0 <v 



TRAVERSE 

TABLE. 





Distance. 

11 Deg. 

11% Deg. 

11% Deg. 

11 % Deg. 

Lat 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 98 

0. 19 

0. 93 

0 . 20 

0. 93 

0 . 

20 

0. 98 | 

0 . 

20 

2 

1. 93 

0. 33 

1. 96 

0. 39 

1. 96 

0 . 

40 

1. 96 

0 . 

41 

3 

2 . 94 

0. 57 

2. 94 

0. 59 

2. 94 

0 . 

60 

2. 94 

0 . 

61 

4 

3. 93 

0. 76 

3. 92 

0. 78 

3. 92 

0 . 

80 

3. 92 

0 . 

82 * 

5 

4. 21 

0. 95 

4. 90 

0. 98 

4. 90 

1 . 

00 

4. 90 

1 . 

02 

G 

5. 83 

1. 14 

5. 88 

1. 17 

5. 88 

1 . 

20 

5. 87 

1 . 

22 

7 

6 . 87 

1. 34 

6 . 87 

1. 37 

6 . 86 

1 . 

40 

6 . 85 

1 . 

43 J 

8 

7. 85 

1. 53 

7. 85 

1. 56 

7. 84 

1 . 

59 

7. 83 

1 . 

63 

9 

8 . 83 

1. 72 

8 . 83 

1. 76 

8 . 82 

1 . 

79 

8. 81 

1 . 

83 

10 

9. 82 

1. 91 

9. 81 

1 95 

9. 80 

1 

99 

9 79 

2. 

C4 

11 

10 . 80 

2 . 10 

10. 79 

2. 15 

10. 78 

2 . 

19 

10 77 

2 . 

24 

12 

11 73 

2 . 20 

11. 77 

2. 34 

11. 76 

2 . 

39 

11 75 

2. 

44 

13 

12. 76 

2. 48 

12. 75 

2. 54 

12. 74 

2 . 

59 

12. 73 

2 . 

65 

14 

13. 74 

2. 67 

13. 73 

2. 73 

13. 72 

2 . 

79 

13. 71 

2 . 

85 

15 

14. 72 

2 . 86 

14. 71 

2. 93 

14. 70 

2 . 

99 

14. 69 

3. 

06 

1 G 

15. 71 

3. 05 

15. 69 

3. 12 

15. 68 

3. 

19 | 

15. 66 

3. 

26 

17 

16. 69 

3. 24 

16. 67 

3. 32 

16. 66 

3. 

39 

16. 64 

3. 

46 

18 

17. 67 

3. 43 

17. 65 

3. 51 

17. 64 

3. 

59 

17. 62 

3. 

66 

19 

18. 65 

3. 63 

18. 63 

3. 71 

18. 62 

3. 

79 

18. 60 

3. 

87 

20 

19 63 

3. 82 

19. 62 

3. 90 

19. 60 

3. 

99 

19. 58 

4. 

07 

21 

20 . 61 

4. 01 

20 . 60 

4. 10 

20. 58 

4. 

19 

20. 56 

4. 

28 

22 

21 . 60 

4. 20 

21. 58 

4. 29 

21. 56 

4. 

39 

21. 54 

4. 

48 

23 

22. 58 

4. 39 

22. 56 

4. 49 

22. 54 

4. 

59 

22, 52 

4. 

68 

24 

23. 56 

4. 53 

23. 54 

4. 68 

23. 52 

4. 

78 

23. 50 

4. 

89 

25 

24. 54 

4. 77 

24. 52 

4. 88 

24. 50 

4. 

98 

24. 48 

5. 

09 

26 

25. 52 

4. 96 

25. 50 

5. 07 

25. 43 

5. 

18 

25. 46 

5. 

80 

27 

26. 50 

5. 15 

26. 48 

5. 27 

26. 46 

5. 

38 

26. 43 

5. 

50 

28 

27. 49 

5. 34 

27. 46 

5. 46 

27. 44 

5. 

58 

27. 41 

5. 

70 

29 

28. 47 

5. 53 

28. 44 

5. 66 

28. 42 

5. 

78 

28. 39 

5. 

91 

30 

29. 45 

5. 72 

29. 42 

5. 85 

29. 40 

5. 

98 

29. 37 

6 . 

11 

35 

34. 36 

6 . 68 

34. 33 

6 . 83 

34. 30 

6 . 

98 

34. 27 

7. 

13 

40 

39. 27 

7. 63 

39. 23 

7. 80 

39. 20 

7. 

97 

39. 16 

8 . 

15 1 

45 

44. 17 

8 . 59 

44. 14 

8 . 78 

44. 10 

8 . 

97 

1 44. 06 

9. 

16 i 

50 

49. 03 

9. 54 

49. 04 

9. 75 

49. 00 

9. 

97 

1 48. 95 

10 . 

18 

55 

53. 99 

10. 49 

53. 94 

10. 73 

53. 90 

10 . 

97 

53. 85 

11 . 

20 1 

i GO 

58. 90 

11. 45 

58. 85 

11. 71 

58. 80 

11 . 

96 

58. 74 

12 . 

22 

G5 

63. 81 

12. 40 

63. 75 

12 . 68 

63. 70 

12 . 

96 

63. 64 

13. 

24 1 

70 

68 . 71 

13. 36 

68 . 66 

13. 66 

68 . 59 

13. 

96 

68 . 53 

14. 

25 j 

75 

! 73. 62 

14. 31 

73. 56 

14. 63 

73. 49 

14. 

95 

73. 43 

15. 

27 ; 

; 80 

78. 53 

15. 26 

78. 46 

15. 61 

78. 39 

15. 

95 

78. 32 

16. 

29 j 

J 85 

83. 44 

16. 22 

83. 37 

16. 53 

83. 29 

16. 

95 

83. 22 

17. 

31 

i 00 

88 . 35 

17. 17 

88 . 27 

17. 56 

88 . 19 

17. 

94 

88 . 11 

18. 

33 

95 

93. 25 

18. 13 

93. 17 

18. 53 

93. 09 

18. 

94 

93. 01 

19. 

35 

100 

98. 16 

19. 08 

98. 08 

19. 51 

97. 99 

19. 

94 

97. 90 

20 . 

36 

6 

O 

G 

cl 

He 

5 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

. Lat. 

79 Deg. 

73% Deg. 

78% Deg. 

73% Deg. 



































































































TRAVERSE TABLE. 83 


a> 

o 

c 

a 

12 Deg. 

12 % 

Deg. 


12 % 

Deg. 


12 % 

Deg 


<n 

5 

Lat. 

Dcp. 

Lat. 

Dcp. 

La 

* 

Dcp. 

Lat. 

Dcp. 

. 1 

0 . 

98 

0 . 

21 

0 . 

93 

0 . 

21 

0 . 

98 

0 . 

22 

0 . 

98 

0 . 

22 

2 

1 . 

96 

0 . 

42 

1 . 

95 

0 . 

42 

1 . 

95 

0 . 

43 

1 . 

95 

0 . 

44 

3 

2 . 

93 

0 . 

62 

2 . 

93 

0 . 

64 

2 . 

93 

0 . 

65 

2 . 

93 

0 . 

06 

4 

3. 

91 

0 . 

83 

3. 

91 

0 . 

85 

3. 

91 

0 . 

87 

3. 

90 

0 . 

88 f 

5 

4. 

89 

1 . 

04 

4. 

89 

1 . 

06 

4. 

88 

1 . 

08 

4. 

88 

1 . 

10 1 

6 

5. 

87 

1 . 

25 

5. 

86 

1 . 

27 

5. 

88 

1 . 

30 

5. 

85 

1 . 

32 } 

7 

6 . 

85 

1 . 

46 

6 . 

84 

1 . 

49 

6 . 

83 

1. 

52 

1 6. 

83 

1 . 

54 

8 

7. 

83 

1 . 

66 1 

7. 

82 

1 . 

70 

7. 

81 

1. 

73 

! 7. 

80 

1 . 

77 

9 

8 . 

80 

1 . 

87 

8 . 

80 

1 . 

91 

8 . 

79 

1 . 

95 

I 8. 

78 

1 . 

99 

10 

9. 

78 

2 . 

03 

9. 

77 

2 . 

12 

9. 

76 

2 . 

18 

! 9. 

75 

2 . 

21 

11 

10 . 

76 

2 . 

29 

10 . 

75 

2 . 

83 

10 . 

74 

2 . 

33 

10 . 

73 

2 . 

43 

12 

11 . 

74 

2 . 

49 

11 . 

73 

2 . 

55 

11 . 

72 

2 . 

60 

1 11. 

70 

2 . 

65 

13 

12 . 

72 

2 . 

70 

| 12. 

70 

2 . 

76 

12 . 

69 

2 . 

81 

12 . 

63 

2 . 

87 

14 

13. 

60 

2 . 

91 

13. 

63 

2 . 

97 

13. 

67 

3. 

03 

13. 

65 

3. 

09 

15 

14. 

67 

3. 

12 

14. 

66 

?. 

13 

14. 

64 

3. 

25 

14. 

63 

3. 

31 

16 

15. 

65 

3. 

33 

15. 

64 

3. 

39 

15. 

62 

3. 

48 

' 15. 

61 

3. 

53 

17 

16. 

63 

3. 

53 

16. 

61 

3. 

61 

13. 

60 

3. 

68 

16. 

53 

3. 

75 

18 

17. 

61 

3. 

74 

17. 

59 

3. 

82 

17. 

57 

3. 

90 

17. 

56 

3. 

97 

19 

18. 

58 

3. 

95 

18. 

57 

4. 

03 

13. 

55 

4. 

11 

18. 

53 

4. 

13 

20 

19. 

56 

4. 

16 

19. 

54 

4. 

24 

19. 

53 

4. 

33 

19. 

51 

4. 

41 

21 

20 . 

54 

4. 

37 

20 . 

52 

4. 

46 

20 . 

50 

4. 

55 

20 . 

48 

4. 

63 

22 

21 . 

52 

4. 

57 

21 . 

50 

4. 

67 

21 . 

43 

4. 

76 

21 . 

46 

4. 

86 

23 

22 . 

50 

4. 

78 

22 . 

43 

4. 

88 

22 . 

45 

4. 

93 

22 . 

43 

5. 

08 

24 

23. 

48 

4. 

99 

23. 

45 

5. 

09 

23. 

43 

5. 

19 

23. 

41 

5. 

30 

25 

24. 

45 

5. 

20 

24. 

43 

5. 

30 

24. 

41 

5. 

41 

24. 

33 

5. 

52 

26 

25. 

43 

5. 

41 

25. 

41 

5. 

52 

25. 

33 

5. 

63 

25. 

36 

5. 

74 

27 

26. 

41 

5. 

61 

26. 

39 

5. 

73 

26. 

86 

5. 

84 

26. 

83 

5. 

96 

23 

27. 

39 

5. 

82 

27. 

36 

5. 

94 

27. 

34 

6 . 

06 

27. 

31 

6 . 

13 

29 

28. 

37 

6 . 

03 

28. 

34 

6 . 

15 

28. 

31 

6 . 

23 

28. 

23 

6 . 

40 

30 

29. 

34 

6 . 

24 

29. 

32 

6 . 

37 

29. 

29 

6 . 

49 

29. 

26 

6 . 

62 

35 

34. 

24 

7. 

23 

34. 

20 

7. 

43 

34. 

17 

7. 

58 

34. 

14 

7. 

72 

40 

39. 

13 

8 . 

32 

39. 

09 

8 . 

49 

39. 

03 

8 . 

66 

39. 

01 

8 . 

83 

45 

44. 

02 

9. 

36 

43. 

93 

9. 

55 

43. 

93 

9. 

74 

43. 

89 

9. 

93 

50 

48. 

91 

10 . 

40 

48. 

86 

10 . 

61 

48. 

81 

10 . 

82 

48. 

77 

11 . 

03 

55 

53. 

80 

11 . 

44 

53. 

75 

11 . 

67 

53. 

70 

11 . 

90 

53. 

64 

12 . 

14 

60 

58. 

69 

12 . 

47 

58. 

63 

12 . 

73 

58. 

58 

12 . 

99 

53. 

52 

13. 

21 

65 

63. 

53 

13. 

51 

63. 

52 

13. 

79 

63. 

46 

14. 

07 

63. 

40 

14. 

35 

70 

68 . 

47 

14. 

55 

68 . 

41 

14. 

85 

68 . 

34 

15. 

15 

63. 

27 

15. 

45 

75 

73. 

36 

15. 

59 

73. 

29 

15. 

91 

73. 

22 

13. 

23 

78. 

15 

16. 

55 

80 

78. 

25 

16. 

63 

78. 

18 

13. 

97 

78. 

10 

17. 

32 

78. 

03 

17. 

66 

85 

83. 

14 

17. 

67 

83. 

06 

18. 

04 

82. 

99 

18. 

40 

82. 

90 

18. 

76 

90 

88 . 

03 

18. 

71 

87. 

95 

19. 

10 

87. 

87 

19. 

48 

87. 

78 

19. 

86 

95 

92. 

92 

19. 

75 

92. 

84 

20 . 

18 

92, 

75 

20 . 

56 

92. 

66 

20 . 

97 

100 

97. 

81 

20 . 

79 

97. 

72 

21 . 

22 

97. 

63 

21 . 

64 

97. 

53 

22 . 

07 

V 

c 

Dcp. 

Lat. 

Dep. 

Lat. 

D( 

p- 

Lat. 

Dcp. 

Lat. 

c5 

cn 

5 

78 Deg. 

7 

7 H 

Deg. 


77 % 

Deg 


77% 

Deg 













































































TRAVERSE TABLE. 


o 

o 

c 

C3 

1 

IS Deo. 

1 3% Deg. 

13% Deg. 

13% Deg. 

00 

1 —< 

Lat. 

Dcp. 

Lat. 

Dcp. 

Lat. 

Dcp. 

Lat. 

Dcp. 

1 

0. 97 

0. 23 

0. 87 

0. 23 

0. 97 

0. 23 

0. 97 

0. 24 

2 

1. 95 

0. 45 

1. 95 

0. 46 

1. 85 

0. 47 

1. 84 

0. 48 

3 

2. 92 

0. 67 

2. 82 

0. 69 

2. 82 

0. 70 

2. 91 

0. 71 

4 

3. 90 

0. 90 

3. 89 

0. 92 

3. 89 

0. 83 

3. 89 

0. 85 

5 

4. 87 

1. 12 

4. 87 

1. 15 

4. 86 

1. 17 

4. 86 

1. 19 

G 

5. 85 

1. 33 

5. 84 

1. 38 

5. 83 

1. 40 

5. 83 

1. 43 

7 

G. 82 

1. 57 

6. 81 

1. 60 

6. 81 

1. 63 

6. 80 

1. 66 

8 

7. 80 

1. 80 

7. 79 

1. 83 

7. 78 

1. 87 1 

7. 77 

1. 80 

9 

8. 77 

2. 02 

8. 76 

2. 06 

8. 75 

2. 10 

8. 74 

2. 14 

19 

9. 74 

2. 25 

9. 73 

2. 29 

9. 72 

2. S3 

9. 71 

2. 38 

11 

10. 73 

2. 47 

10. 71 

2. o2 

10. 70 

2. 57 

10. 68 

2. 61 

13 

11. GO 

2. 70 j 

11. 63 

2. 75 

11. 67 

2. 80 

11. 66 

2. 85 

13 

12. G7 

2. 82 

12. 63 

2. 98 

12. 64 

3. 03 

1 12. 63 

3. 09 

11 

13. 64 

3. 15 

13. 63 

3. 21 

13. 61 

3. 27 

13. 60 

3. 33 

13 

14. 62 

3. 37 

14. 60 

3. 44 

14. 59 

3. 50 

14. 57 

3. 57 

1G 

15. 59 

3. 60 

13. 57 

3. 67 

15. 56 

3. 74 

15. 54 

3. 80 

17 

16. 57 

3. 82 

16. 55 

3. 80 

16. 53 

3. 97 

16. 51 

4. 04 

18 

17. 54 

4. 05 

17. 52 

4. 13 

17. 50 

4. 20 

17. 48 

4. 28 

19 

18. 51 

4. 27 

18. 49 

4. 35 

1 18. 48 

4. 44 

18. 4G 

4. 52 

20 

19. 49 

4. 50 

19. 47 

4. 58 

I 19. 45 

4. 67 

19. 43 

4. 75 

21 

20. 46 

4. 72 

20. 44 

4. 81 

20. 42 

4. 90 

20. 40 

4. 99 

22 

21. 44 

4. 85 

21. 41 

5. 04 

21. 39 

5. 14 

21. 37 

5. 23 

23 

22. 41 

5. 17 

22. 39 

5. 27 

22. 36 

5. 37 

22. 34 

5. 47 

24 

23. 38 

5. 40 

23. 86 

5. 50 

23. 34 

5. 60 

23. 31 

5. 70 

25 

24. 36 

5. 62 

24. 33 

5. 73 

24. 81 

5. 84 

24. 28 

5. 84 

26 

25. 33 

5. 85 

25. 31 

5. 96 

25. 28 

6. 07 

25. 25 

6. 18 

27 

26. 31 

6. 07 

26. 23 

6. 19 

26. 25 

6. 30 

26. 23 

6. 42 

23 

27. 23 

6. 30 

27. 25 

6. 42 

27. 23 

6. 54 

27. 20 

6. 66 

29 

23. 23 

6. 52 

23. 23 

6. 65 

j 28. 20 

6. 77 

28. 17 

6. 89 

30 

29. 23 

6. 75 

29. 20 

6. 88 

29. 17 

7. 00 

29. 14 

7. 13 

33 

34. 10 

7. 87 

34. 07 

8. 02 

84. 03 

8. 17 

34. 00 

8. 32 

40 

33. 97 

9. 00 

38. 84 

9. 17 

38. 89 

9. 34 

88. 85 

9. 51 

45 

43. 85 

10. 12 

43. 80 

10. 81 

43. 76 

10. 51 

43. 71 

10. 70 

50 

43. 72 

11. 25 

48. 67 

11. 46 

48. 62 

11. 67 

48. 57 

11. 88 

53 

53. 59 

12. 37 

53. 54 

12. 61 

53. 48 

12. 84 

53. 42 

13. 07 

GO 

58. 46 

13. 50 

58. 40 

13. 75 

58. 34 

14. 01 

58. 28 

14. 26 

63 

63. 33 

14. 62 

63. 27 

14. 80 

63. 20 

15. 17 

63. 14 

15. 45 

70 

68. 21 

15. 75 

68. 14 

16. 04 

68. 07 

16. 34 

67. 89 

16. 64 

75 

73. 08 

16. 87 

73. 00 

17. 19 

72. 93 

17. 50 

72. 85 

17. 83 

80 

77. 95 

18. 00 

77. 87 

18. 34 

77. 79 

18. 68 

77. 71 

19. 01 

85 

82. 82 

19. 12 

82. 74 

19. 48 

82. 65 

19. 84 

82. 56 

20. 20 

80 

87. 69 

20. 25 

87. 60 

20. 63 

87. 51 

21. 01 

87. 42 

21. 89 

95 

82. 57 

21. 37 

92. 47 

21. 77 

92. 38 

22. 18 

92. 28 

22. 58 

100 

97. 44 

22. 50 

97. 34 

22. 82 

87. 24 

23. 34 

97. 13 

23. 77 

6 

O 

c 

Dcp. 

Lat. 

Dop. 

Lat. 

Dcp. 

Lat. 

Dop. 

Lat 

cl 

n 

s 

77 Deo. 

70% Deg. 

76% Deg. 

76% Deg. 




























































































TRAVERSE 

TABLE. 





85 

1 

8 & 

9 a 

1 » 

5 

14 Deg. 

14% Deg. 

14 % Deg. 

14% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

j Dep. 

j Lat. 

1 

Dep. 

1 

0. 97 

0. 24 

0. 97 

0. 25 

0. 97 

0. 

25 

0. 

97 

0. 

25 

2 

1. 94 

0. 48 

1. 94 

0. 49 

1. 94 

0. 

50 

i. 

93 

0. 

51 

3 

2. 91 

0. 73 

2. 91 

0. 74 

2. 90 

0. 

75 

2. 

90 

0. 

76 

4 

3. 83 

0. 97 

3. 88 

0. 98 

3. 87 

1. 

00 

3. 

87 

1. 

02 

5 

4. 85 

1. 21 

4. 85 

1. 23 

4. 84 

1. 

25 

4. 

84 

1. 

27 

G 

5. 82 

1. 45 

5. 82 

1. 43 

5. 81 

1. 

50 

5. 

80 

1. 

53 

7 

6. 79 

1. 69 

6. 78 

1. 72 

6. 78 

1. 

75 

6. 

77 

1. 

78 

8 

7. 76 

1. 94 

7. 75 

1. 97 

7. 75 

2. 

00 

7. 

74 

2. 

04 

9 

8. 73 

2. 18 

8. 72 

2. 22 

8. 71 

2. 

25 

8. 

70 

2. 

29 

10 

9. 70 

2. 42 

9. 69 

2. 46 

9. 63 

2. 

50 

9. 

67 

2t 

55 

11 

10. 67 

2. 66 

10. 66 

2. 71 

10. 65 

2. 

75 

10. 

64 

2. 

80 

12 

11. 64 

2. 90 

11. 63 

2. 95 

11. 62 

3. 

00 

11. 

60 

3. 

06 

1 13 

12. 61 

3. 15 

12. 60 

3. 20 

12. 59 

3. 

25 

12. 

57 

3. 

31 

1 14 

13. 53 

3. 39 

13. 57 

3. 45 

13. 55 

3. 

51 

13. 

54 

3. 

56 

15 

14. 55 

3. 63 

14. 54 

3. 69 

14. 52 

3. 

76 

14. 

51 

3. 

82 

1 10 

15. 52 

3. 87 

15. 51 

3. 94 

15. 49 

4. 

01 

15. 

47 

4. 

07 

17 

16. 50 

4. 11 

16. 43 

4. 18 

16. 46 

4. 

26 

16. 

44 

4. 

33 

18 

17. 47 

4. 35 

17. 45 

4. 43 

17. 43 

4. 

51 

17. 

41 

4. 

58 

1 19 

18. 44 

4. 60 

18. 42 

4. 68 

18. 39 

4. 

76 

18. 

37 

4. 

84 

I 20 

19. 41 

4. 84 

19. 38 

4. 92 

19. 36 

5. 

01 

19. 

34 

5. 

09 

21 

20. 38 

5. 03 

20. 35 

5. 17 

20. 33 

5. 

26 

20. 

31 

5. 

35 

j 22 

21. 35 

5. 32 

21. 32 

5. 42 

21. 30 

5. 

51 

21. 

28 

5. 

60 

23 

22. 32 

5. 56 

22. 29 

5. 66 

22. 27 

5. 

76 

22. 

24 

5. 

86 

1 24 

23. 29 

5. 81 

23. 26 

5. 91 

23. 24 

6. 

01 

23. 

21 

6. 

11 

25 

24. 26 

6. 05 

24. 23 

6. 15 

24. 20 

6. 

26 

24. 

18 

6. 

37 

1 26 

25. 23 

6. 29 

25. 20 

6. 40 

25. 17 

6. 

51 

25. 

14 

6. 

62 

27 

26. 20 

6. 53 

26. 17 

6. 65 

26. 14 

6. 

76 

26. 

11 

6. 

87 

I 28 

27. 17 

6. 77 

27. 14 

6. 89 

27. 11 

7. 

01 

27. 

08 

7. 

13 

j 29 

28. 14 

7. 02 

28. 11 

7. 14 

28. 08 

7. 

26 

28. 

04 

7. 

38 

30 

29. 11 

7. 26 

29. 08 

7. 38 

29. 04 

7. 

51 

29. 

01 

7. 

64 

1 3 5 

33. 96 

8. 47 

33. 92 

8. 62 

33. 89 

8. 

76 

33. 

85 

8. 

91 

40 

38. 81 

9. 68 

38. 77 ! 

9. 85 

38. 73 

10. 

02 

88. 

68 

10. 

18 

45 

43. 66 

10. 89 

43. 62 

11. 08 

43. 57 

11. 

27 

43. 

52 

11. 

46 

50 

48. 51 

12. 10 

48. 46 

12. 31 

48. 41 

12. 

52 

48. 

35 

12. 

73 

| 55 

53. 37 

13. 31 

53. 31 

13. 54 

53. 25 

13. 

77 

j 53. 

19 

14. 

00 

60 

58. 22 

14. 52 

58. 13 

14. 77 

58. 09 

15. 

02 

58. 

02 

15. 

28 

65 

63. 07 

15. 72 

63. 00 

16. 00 

62. 93 

16. 

27 

62. 

86 

16. 

55 

70 

67. 92 

16. 93 

67. 85 ! 

17. 23 

67. 77 

17. 

53 

67. 

69 

17, 

82 

75 

72. 77 

18. 14 

72. 69 

18. 46 

72. 61 

18. 

78 

72. 

53 

19. 

10 

80 

77. 62 

19. 35 

77. 54 

19. 69 

77. 45 

20. 

03 

77. 

36 

20. 

37 

85 

82. 48 

20. 56 

82. 38 

20. 92 

82. 29 | 

21. 

28 

82. 

20 

21. 

64 

90 

87. 33 

21. 77 

87. 23 

22. 15 

87. 13 

22. 

53 

87. 

03 

22. 

91 

95 

92. 18 

22. 98 

92. 08 

23. 38 

91. 97 

23. 

79 

91. 

87 

24. 

19 

100 

97. 03 

24. 19 

96. 92 

24. 62 

96. 81 

25. 

04 

96. 

70 

25. 

46 

i j 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

76 Deg. 

75 % Deg. 

75% Deg. 

75% Deg. 






































































































86 TRAVERSE TABLE. 


! 

o 

o 

c 

15 Deo. 

15% Deg. 

15X Deg. 

15% Deg. 

/j 

5 

Lat. 

Dep. j 

L .t. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 97 

0. 26 1 

0. 90 I 

0. 20 

0. 90 

0. 27 

0. 96 

0. 27 

2 

1. 93 

0. 52 

1. 93 

0. 53 

1. 03 

0. 53 

1. 92 

0. 54 

3 

2. 90 

0. 73 

2. 89 

0. 79 

2. 89 

0 . so 

2. 89 

0. 81 

4 

3. 60 

1. Cl 

3. 80 

1. (' 5 

3. 85 

1. 07 

3. 85 

1. C9 

5 

4. 83 

1. 2D 

4. 82 

1. CD 

4. 82 

1. 34 

4. 81 

1. 36 ' 

G 

5. 80 

1. 55 

5. 79 

1. 53 

5. 73 

1. (0 

5. 77 

1. C3 

7 

6 . 70 

1. 61 

0. 75 

1. 84 

6. 75 

1. 87 

6. 74 

1. 90 

8 

7. 73 

2. o; 

7. 72 

2. 10 

7. 71 

2. 11 

7. 70 

2. 17 

9 

8. CD 

o Ol 

KJO 

8. 03 

2. C7 

8. 67 

2. 41 

8. CG 

2. 41 

10 

9. GO 

2. 59 

9. 65 

2. 63 

9. 61 

2. 67 

9. 62 

2. 71 

11 

10. G3 

2. 85 

ID. 61 

2. 89 

10. CO 

2. 04 

10. 59 

2. 99 

12 

11. 59 

3. 11 

11. 53 

3. 13 

11. 53 

3. 21 

11. 55 

3. 26 

13 

12. 50 

O 

u. OO 

12. 51 

3. 42 

12. 53 

3. 47 

12. 51 

3. 53 

14 

13. 52 

3. G3 

13. 51 

3. 63 

13. 49 

3. 74 

13. 47 

3. 80 

15 

14. 49 

3. 83 

11. 47 

3. 95 

14. 45 

4. 01 

14. 41 

4. 07 

10 

13. 4 > 

4. 11 

15. 41 

4. 21 

15. 43 

4. 28 

15. 40 

4. 34 

17 

10. 42 

4. 40 

13. 40 

4. 47 

16. 33 

4. 54 

10. 36 

4. 61 

18 

17. 3D 

4. GG 

17. 37 

4. 73 

17. 35 

4. 81 

17. 32 

4. 89 

ID 

13. 35 

4. 92 

13. 33 

5. 00 

13. 31 

5. 03 

18. 29 

5. 16 

20 

ID. 3J 

5. 13 

19. 30 

5. 20 

19. 27 

5. 34 

19. 25 

5. 43 

21 

20. 23 

5. 41 

20. 26 

5. 52 

20. 24 

5. 61 

20. 21 

5. 70 

oo 

21. 25 

5. 09 

21. 23 

5. 79 

21. 20 

5. 83 

21. 17 

5. 97 

23 

22. 22 

5. 95 

22. ID 

6. 05 

22. 10 

6. 15 

22. 14 

6. 24 

24 

23. 18 

G. 21 

23. 15 

6. 31 

23. 13 

6. 41 

23. 10 

6. 51 

25 

24. 15 

G. 47 

21. 12 

6. 58 

24. 09 

6. 63 

24. 06 

6. 79 

20 

25. 11 

6. 73 

25. 03 

6. 84 

25. 05 

6. 95 

25. 02 

7. 06 

27 

26. 03 

6. 99 

2G. C5 

7. 10 

20. 02 

7. 22 

25. 99 

7. 33 

28 

27. 05 

7. 25 

27. 01 

7. 33 

20. 93 

7. 43 

26. 95 

7. 60 

29 

28. 01 

7. 51 

27. 98 

7. 63 

27. 95 

7. 75 

27. 91 

7. 87 

30 

28. 98 

7. 76 

28. 94 

7. 89 

28. 91 

8. 02 

28. 87 

8. 14 

35 

33. 81 

9. C6 

83. 77 

9. 21 

33. 73 

9. 35 

33. 69 

9. 50 

40 

38. G4 

10. 35 

38. 59 

10. 52 

38. 55 

10. 69 

38. 50 

10. 86 

45 

43. 47 

11. 65 

43. 42 

11. 84 

43. 3G 

12. 03 

43. 31 

12. 21 

50 

48. 30 

12. 94 

48. 24 

13. 15 

48. 18 

13. 36 

48. 12 

13. 57 

55 

53. 13 

14. 24 

53. 00 

14. 47 

53. 00 

14. 70 

52. 94 

14. 93 

GO 

57. 90 

15. 53 

57. 89 

15. 78 

57. 82 

16. 03 

57. 75 

16. 29 

G5 

G2. 79 

16. 82 

G2. 71 

17. 10 

62. 64 

17. 37 

62. 56 

17. 64 

70 

G7. G1 

18. 12 

G7. 54 

18. 41 

67. 45 

18. 71 

67. 37 

19. 00 

75 

72. 44 

19. 41 

72. 36 

19. 73 

72. 27 

20. 04 

72. 18 

20. 36 

80 

77. 27 

20. 71 

77. 18 

21. 04 

77. 09 

21. 38 

77. 00 

21. 72 

85 

82. 10 

22. 00 

82. 01 

22. 36 

81. 91 

22. 72 

81. 81 

23. 07 

90 

86. 93 

23. 29 

86. 83 

23. 67 

86. 73 

24. 05 

86. 62 

24. 43 

95 

91. 76 

24. 59 

91. G5 

24. 99 

91. 54 

25. 39 

91. 43 

25. 79 

100 

96. 59 

25. 88 

96. 48 

26. 30 

96. 36 

j 26. 72 

96. 25 

27. 14 

6 

V 

n 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

a 

<-> 

\ .2 

' P 

75 

Deg. 

74% Deg. 

74% Deg. 

74% Deg. 


- w - 


























































































TRAVERSE TABLE. 87 


6 

V 

G 

c3 

16 Deo. 

16% 

Deg 


! 16% 

Deg 


16% 

i 

Deg 


cn 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lit. 

Dep. 

1 

0 . 

96 

0 . 

28 

0 . 

96 

0 . 

28 

0 . 

96 

0 . 

28 

0 . 

96 

0 . 

29 

2 

1 . 

9.3 

0 . 

55 

1 . 

92 

0 . 

56 

1 . 

92 

0. 

57 

i. 

92 

0 . 

58 

3 

2. 

83 

0 . 

83 

2. 

83 

0 . 

84 

2. 

88 

0 . 

85 

2. 

87 

0 . 

86 

4 

3. 

85 

1 . 

10 

3. 

84 

1 . 

12 

3. 

84 

1 . 

14 

3. 

83 

1 . 

15 

5 

4. 

81 

1 . 

33 

4. 

80 

1 . 

40 

4. 

79 

1 . 

42 

4. 

79 

1 . 

44 

G 

5. 

77 

1 . 

G5 

5. 

76 

1. 

63 

5. 

75 

1 . 

70 

5. 

75 

1 . 

73 

7 

G. 

r* > 
i J 

1 . 

93 

6. 

72 

1. 

96 

6. 

71 

1 . 

99 

6. 

70 

2. 

02 

8 

7. 

GO 

2. 

21 

7. 

63 

2. 

24 

7. 

67 

2. 

27 

7. 

66 

2. 

31 

9 

8. 

Gj 

2. 

43 

8. 

61 

2. 

52 

8. 

63 

2. 

56 

8. 

62 

2. 

50 

10 

9. 

Cl 

2. 

76 

9. 

60 

2. 

80 

9. 

59 

2. 

84 

9. 

53 

2. 

83 

li 

10. 

57 

3. 

03 

10. 

53 

3. 

03 

10. 

55 

3. 

12 

10. 

53 

3. 

17 

13 

11. 

51 

3. 

31 

11. 

52 

3. 

36 

n. 

51 

3. 

41 

11. 

49 

3. 

46 

13 

12 . 

50 

3. 

53 

12. 

4 3 

3. 

64 

I 13. 

4G 

3. 

G9 

12. 

45 

3. 

75 

14 

13. 

46 

3. 

86 

13. 

41 

3. 

92 

I 13. 

43 

3. 

93 

13. 

41 

4. 

03 

15 

14. 

43 

4. 

13 

14. 

40 

4. 

20 

14. 

S3 

4. 

26 

14. 

Ci o 

C'O 

4. 

32 

1G 

15. 

33 

4. 

41 

15. 

Qfr 

Ol) 

4. 

4 3 

1 15. 

31 

4. 

54 

15. 

32 

4. 

61 * 

17 

1G. 

34 

4. 

60 

16. 

3) 

4. 

7 6 

36. 

30 

4. 

63 

16. 

28 

4. 

90 

18 

17. 

80 

4. 

96 

17. 

23 

5. 

04 

17. 

26 

5. 

11 

17. 

24 

5. 

19 

19 

18. 

26 

5. 

24 

13. 

21 

5. 

S3 

18. 

23 

5. 

40 

18. 

10 

5. 

48 

20 

19. 

23 

5. 

51 

19. 

20 

5. 

60 

19. 

18 

5. 

68 

19. 

13 

5. 

76 

21 

20. 

19 

5. 

79 

20. 

16 

5. 

88 

20. 

14 

5. 

96 

20. 

11 

6. 

05 

22 

21. 

15 

6 . 

06 

21. 

13 

6. 

16 

21. 

00 

6. 

25 

21. 

07 

6. 

34 

23 

22. 

11 

6 . 

34 

22. 

03 

6. 

44 

22. 

05 

6. 

53 

22. 

02 

6. 

C3 

24 

23. 

07 

G. 

62 

23. 

01 

6. 

72 

23. 

01 

6. 

82 

22. 

98 

6. 

92 

25 

24. 

03 

G. 

89 

24. 

CO 

7. 

00 

23. 

97 

7. 

10 

23. 

94 

7. 

20 

26 

24. 

99 

7. 

17 

24. 

96 

7. 

23 

24. 

93 

7. 

33 

24. 

90 

7. 

49 

27 

25. 

95 

7. 

44 

25. 

92 

7. 

56 

25. 

89 

7. 

67 

25. 

85 

7. 

78 

28 

2G. 

93 

7. 

72 

26. 

83 

7. 

81 

26. 

85 

7. 

95 

26. 

81 

8. 

07 

29 

27. 

83 

7. 

99 

•27. 

81 

8. 

11 

27. 

81 

8. 

24 

27. 

77 

8. 

36 

30 

28. 

81 

8. 

27 

28. 

80 

8. 

39 

28. 

76 

8. 

52 

28. 

73 

8. 

65 

35 

33. 

G4 

9. 

65 

33. 

60 

9. 

79 

33. 

56 

9. 

94 

33. 

51 

10. 

09 

40 

38. 

45 

11. 

03 

38. 

40 

11. 

19 

38. 

85 

11. 

36 

38. 

SO 

11. 

53 

45 

43. 

2G 

12. 

40 

43. 

20 

12. 

59 

43. 

15 

12. 

78 

43. 

09 

12. 

97 

50 

48. 

06 

13. 

78 

48. 

00 

13. 

99 

47. 

94 

14. 

20 

47. 

88 

14. 

41 

55 

52. 

87 

15. 

16 

52. 

80 

15. 

39 

52. 

74 

15. 

62 

52. 

67 

15. 

85 

GO 

57. 

68 

16. 

54 

57. 

60 

16. 

79 

57. 

53 

17. 

04 

57. 

45 

17. 

29 

G5 

G2. 

48 

17. 

92 

62. 

40 

18. 

19 

62. 

32 

18. 

46 

62. 

24 

18. 

73 

70 

67. 

29 

19. 

29 

67. 

20 

19. 

59 

67. 

12 

19. 

88 

67. 

03 

20. 

17 

75 

72. 

09 

20. 

67 

72. 

00 

20. 

99 

71. 

91 

21. 

30 

71. 

82 

21. 

61 

80 

76. 

90 

22. 

05 

76. 

80 

22. 

39 

76. 

71 

22. 

72 

76. 

61 

23. 

06 

85 

81. 

71 

23. 

43 

81. 

60 

23. 

79 

81. 

50 

24. 

14 

81. 

39 

24. 

50 

90 

86. 

51 

24. 

81 

86. 

40 

25. 

18 

86. 

29 

25. 

56 

86. 

18 

25. 

94 

95 

91. 

32 

26. 

19 

91. 

20 

26. 

58 

91. 

09 

26. 

98 

90. 

97 

27. 

88 

100 

96. 

13 

27. 

56 

96. 

00 

27. 

98 

95. 

88 

28. 

40 

95. 

76 

28. 

82 

V 

O 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

eS 

■si 

6 

74 Deg. 

7SK 

Deg. 


i 

3% 

Deg. 


73% 

Dl G 


















































































88 


TRAVERSE 

TABLE. 



9 

c 

c3 

17 Deg. 

17H 

Deg. 

17% 

Deg. 

17% 

Deg. 

S to 

£ 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

| 

Lat. 

Dep. 

1 

0. 96 

0. 29 

0. 95 

0. 30 

0. 95 | 

0. 30 

0. 95 

0. 30 

2 

1. 91 

0. 58 

1. 91 

0. 59 

1. 91 

0. 60 

1. 90 

0. 61 

3 

2. 87 

0. 83 

2. 87 

0. 89 

2. 86 

0. 90 i 

2. 86 

0. 91 

4 

3. 83 

1. 17 1 

3. 82 

1. 19 

3. 81 

1. 20 

3. 81 

1. 22 

5 

4. 78 

1. 46 | 

4. 78 

1. 48 

4. 77 

1. 50 

4. 76 

1. 52 | 

6 

5. 74 

1. 75 | 

5. 73 

1. 78 

5. 72 

1. 80 

5. 71 

1. 83 

7 

6. 69 

2. 05 

6. 69 

2. 08 

6. 63 

2. 10 

6. 67 

2. 13 

8 

7. 65 

2. 34 

7. 64 

2. 37 

7. 63 

2. 41 

7. 62 

2. 44 

9 

8. 61 

2. 63 

8. 60 

2. 67 

8. 58 

2. 71 

8. 57 

2. 74 

10 

9. 56 

2. 92 

9. 55 

2. 97 

9. 54 

3. 01 

9. 52 

3. 05 

11 

10. 52 

3. 22 

10. 51 

3. 26 

10. 49 

3. 31 

10. 43 

3. 35 

12 

11. 43 

3. 51 

11. 43 

3. 56 

11. 44 

3. 61 

11. 43 

3. 66 

13 

12. 43 

3. 80 

12. 42 

3. 85 

12. 40 

3. 91 1 

12. 38 

3. 96 

14 

13. 39 

4. 09 

13. 37 

4. 15 

13. 35 

4. 21 

13. 33 

4. 27 

15 

14. 34 

4. 39 

14. 33 

4. 45 

14. 31 

4. 51 

14. 29 

4. 57 

10 

15. 30 

4. 63 

15. 28 

4. 74 

15. 26 

4. 81 

15. 24 

4. 88 

17 

16. 26 

4. 97 

16. 24 

5. 04 

16. 21 

5. 11 

16. 19 

5. 18 

18 

17. 21 

5. 26 

17. 19 

5. 34 

17. 17 

5. 41 

17. 14 

5. 49 

19 

18. 17 

5. 56 1 

18. 15 

5. 63 

18. 12 

5. 71 

18. 10 

5. 79 

20 

19. 13 

5. 85 | 

19. 10 

5. 93 

19. 07 

6. 01 

19. 05 

6. 10 

21 

20. 08 

6. 14 | 

20. 06 

6. 23 

20. 03 

6. 31 

20. 00 

6. 40 

22 

21. 04 

6. 43 

21. 01 

6. 52 

20. 98 

6. 62 

20. 95 

6. 71 

23 

21. 99 

6. 72 

21. 97 

6. 82 

21. 94 

6. 92 

21. 91 

7. 01 

24 

22. 95 

7. 02 

22. 92 

7. 12 

22. 89 

7. 22 

22. 86 

7. 32 

25 

23. 91 

7. 31 

23. 88 

7. 41 

23. 84 

7. 52 

23. 81 

7. 62 

26 

24. 86 

7. 60 

24. 83 

7. 71 

24. 80 

7. 82 

24. 76 

7. 93 

27 

25. 82 

7. 89 

25. 79 

8. 01 

25. 75 

8. 12 

25. 71 

8. 23 

28 

26. 78 

8. 19 

26. 74 

8. 30 

26. 70 

8. 42 

26. 67 

8. 54 

29 

27. 73 

8. 48 

27. 70 

8. 60 

27. 66 

8. 72 

27. 62 

8. 84 

30 

28. 69 

8. 77 

28. 65 

8. 90 

28. 61 

9. 02 

28. 57 

9. 15 

35 

33. 47 

10. 23 

33. 43 

10. 38 

| 33. 38 

10. 52 

33. 33 

10. 67 

40 

38. 25 

11. 69 

38. 20 

11. 86 

38. 15 

12. 03 

38. 10 

12. 19 

45 

43. 03 

13. 16 

42. 98 

13. 34 

42. 92 

13. 53 

42. 86 

13. 72 

50 

47. 82 

14. 62 

47. 75 

14. 83 

47. 69 

15. 04 

47. 62 

15. 24 

55 

52. 60 

16. 08 

52. 53 

16. 31 

52. 45 

16. 54 

52. 38 

16. 77 

60 

57. 38 

17. 54 

57. 30 

17. 79 

57. 22 

18. 04 

57. 14 

18. 29 

65 

62. 16 

19. 00 

62. 08 

19. 28 

61. 99 

19. 55 

61. 91 

19. 82 

70 

66. 94 

20. 47 

66. 85 

20. 76 

66. 76 

21. 05 

66. 67 

21. 34 

75 

71. 72 

21. 93 

71. 63 

22. 24 

71. 53 

22. 55 

71. 43 

22. 86 

80 

76. 50 

23. 39 

76. 40 

23. 72 

76. 30 

24. 06 

76. 19 

24. 39 

85 

81. 29 

24. 85 

81. 18 

25. 21 

81. 07 

25. 56 

80. 95 

25. 91 

90 

86. 07 

26. 31 

85. 95 

26. 69 

85. 83 

27. 06 

85. 72 

27. 44 

1 95 

90. 85 

27. 78 

90. 73 

28. 17 

90. 60 

23. 57 

90. 48 

28. 96 

100 

95. G3 

29. 24 

95. 50 

1 29. 65 

1 

95. 37 

30. 07 

95. 24 

30. 49 

6 

\ o 

Dep. 

Lat.‘ 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

cl 

1 5 

73 

Deg. 

72% Deg. 

WA 

Deg. 

72% Deg. 
































































































I 



TRAVERSE 

TABLE. 


89 

Distance. 

18 Deg. 

18% Deg. 

18% Deg. 

18% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 Lat 

Dep. 

1 

0. 95 

0 . 31 

0. 95 

0. 31 

0. 95 

0. 32 

0. 95 

0. 32 

2 

1. 90 

0 . 62 

1. 90 

0. 63 

1 . 90 

0. 63 

1 1. 89 

0. 64 | 

3 

2. 85 

0. 93 

2. 85 

0. 94 

2. 84 

0. 95 

2 . 84 

0. 96 | 

4 

3. 80 

1. 24 

3. 80 

1. 25 

3. 79 

1. 27 

3. 79 

1. 29 * 

5 

4. 76 

1. 55 

4. 75 

1. 57 

4. 74 

1. 59 

| 4. 73 

1 . 61 

G 

5. 71 

1. 85 

5. 70 

1 . 88 

5. 69 

1. 90 

5. 68 

1. 93 

7 

6 . 66 

2 . 16 

6 . 65 

2. 19 

6 . 64 

2 . 22 

6 . 63 

2. 25 

8 

7. 61 

2. 47 

7. 60 

2. 51 

7. 59 

2. 54 

7. 58 

2, 57 

9 

8 . 56 

2. 78 

8 . 55 

2 . 82 

8 . 53 

2 . 86 

! 8. 52 

2. 89 

10 

9. 51 

3. 09 

9. 50 

3. 13 

9. 48 

3. 17 

1 9. 47 

3. 21 

11 

10. 46 

3. 40 

10 . 45 

3. 44 

10. 43 

3. 49 

10. 42 

3. 54 j 

12 

11. 41 

3. 71 

11 . 40 

| 3. 76 

11. 33 

3. 81 

11. 36 

3. 86 i 

13 

12. 36 

4. 02 

12. 35 

4. 07 

12. 33 

4. 12 

12. 31 

4. 18 

14 

13. 31 

4. 33 

13. 30 

4. 38 

13. 28 

4. 44 

13. 26 

4. 50 

1 15 

14. 27 

4. 64 

14. 25 

4. 70 

I 14. 22 

4. 76 

14. 20 

4. 82 

( 18 

15. 22 

4. 94 

15. 20 

5. 01 

j 15. 17 

5. 08 

15. 15 

5. 14 

17 

16. 17 

5. 25 

16. 14 

5. 32 

1 16. 12 

5. 39 

16. 10 

5. 46 

18 

17. 12 

5. 56 

17. 09 

5. 64 

17. 07 

5. 71 

17. 04 

5. 79 

19 

18. 07 

5. 87 

18. 04 

5. 95 

18. 02 

6 . 03 

17. 99 

6 . 11 

( 20 

19. 02 

6 . 18 

18. 99 

6 . 26 

18. 97 

6 . 35 

18. 94 

6 . 43 1 

' 21 

19. 97 

6 . 49 

19. 94 

6 . 58 

19. 91 

6 . 66 

19. 89 

6 . 75 j 

' 22 

20. 92 

6 . 80 

1 20. 89 

6 . 89 

I 20 . 86 

6 . 98 

20. 83 

7. 07 

( 23 

21. 87 

7. 11 

21. 84 

7. 20 

j 21 . 81 

7. 30 

21 . 78 

7. 39 j 

; 24 

22. 83 

7. 42 

22. 79 

7. 52 

1 22. 76 

7. 62 

22. 73 

7. 71 | 

f 25 

23. 78 

7. 73 

23. 74 

7. 83 

| 23. 71 

7. 93 

23. 67 

8 . 04 i 

26 

24. 73 

8 . 03 

24. 69 

8 . 14 

24. 66 

8 . 25 

24. 62 

8 . 36 i 

1 27 

25. 68 

8 . 34 

25. 64 

8 . 46 

25. 60 

8 . 57 

25. 57 

8 . 68 

28 

26. 63 

8 . 65 

26. 59 

8 . 77 

26. 55 

8 . 88 

26. 51 

9. 00 

f 29 , 

27. 58 

8 . 96 

27. 54 

9. 08 

27. 50 

9. 20 

27. 46 

9. 32 

1 30 

28. 53 

9. 27 

28. 49 

9. 39 

28. 45 

9. 52 

28. 41 

9. 64 

1 . 35 

33. 29 

10 . 82 

33. 24 

10. 96 

33. 19 

11 . 11 

33. 14 

11. 25 

f 40 

38. 04 

12 . 36 

37. 99 

12. 53 

37. 93 

12. 69 

37. 88 

12 . 86 

( 45 

42. 80 

13. 91 

42. 74 

14. 09 

42. 67 

14. 28 

42. 61 

14. 46 

50 

47. 55 

15. 45 

47. 48 

15. 66 

47. 42 

15. 87 

47. 35 

16. 07 

1 55 

52. 31 

17. 00 

52. 23 

17. 22 

52. 16 

17. 45 

52. 08 

17. 68 

( GO 

57. 06 

18. 54 

56. 98 

18. 79 

56. 90 

19. 04 

56. 82 

19. 29 

f 65 

61. 82 

20. 09 

61. 73 

20. 36 

61. 64 

20 . 62 

61. 55 

20. 89 | 

70 

66 . 57 

21. 63 

66 . 48 

21. 92 

66 . 38 

22 . 21 

66 . 29 

22. 50 i 

75 

71. 33 

23. 18 

71. 23 

23. 49 

71. 12 

23. 80 

71. 02 

24. 11 

80 

76. 08 

24. 72 

75. 98 

25. 05 

75. 87 

25. 38 

75. 75 

25. 72 

85 

80. 84 

26. 27 

80. 72 

26. 62 

80. 61 

26. 97 

80. 49 

27. 32 

90 

85. 60 

27. 81 

85. 47 

28. 18 

85. 35 

28. 56 

85. 22 

28. 93 

95 

90. 35 

29. 36 

90. 22- 

29. 75 

90. 09 

30. 14 

89. 96 

30. 54 

100 

95. 11 

so. to ; 

94. 97 

31. 32 

94. 83 

31. 73 

94. 69 

32. 14 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

72 Deg. 

71% Deg. 

71% Deo. 

71% Deg. 


5 






















































































































90 



TRAVERSE 

TABLE. 



6 

o 

a 

Sj 

19 Deg. 

19% Deg. 

19% Deg. 

19% Deg. 

/} 

ft 

Lat. 

Bop. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 95 

0. 33 

0. 94 

0. 33 

0. 94 

0. 33 

0. 94 

0. 34 

2 

1. 89 

0. 63 

1. 89 

0. 66 

1. 89 

0. 67 

1. 88 

0. 68 

3 

2. 84 

0. 98 

2. 83 

0. 99 

2. 83 

1. 00 

2. 82 

1. 01 

4 

3. 73 

1. 30 

3. 78 

1. 32 

3. 77 

1. 34 

3. 76 

1. 35 

5 

4. 73 

1. 63 

4. 72 

1. 65 

4. 71 

1. 67 

4. 71 

1. G9 

6 

5. G7 

1. 95 

5. 63 

1. 98 

5. 66 

2. 00 

5. G5 

2. 03 

7 

G. 62 

2. 28 

6. 61 

2. 31 

6. 60 

2. 34 

6. 59 

2. 37 

8 

7. 56 

2. GO 

7. 55 

2. 64 

7. 54 

2. 67 

7. 53 

2. 70 

9 

8. 51 

2. 93 

8. 50 

2. 97 

8. 48 

3. 00 

8. 47 

3. 04 

10 

9. 46 

3. 26 

9. 44 

3. 30 

9. 43 

3. 34 

9. 41 

3. 38 

11 

10. 40 

3. 58 

10. 38 

3. 63 

10. 37 

3. 67 

10. 35 

3. 72 

12 

11. 35 

3. 91 

11. 33 

3. 96 

11. 31 

4. 01 

11. 29 

4. 06 

13 

12. 29 

4. 23 

12. 27 

4. 29 

12. 25 

4. 34 

12. 24 

4. 39 

14 

13. 24 

4. 56 

13. 22 

4. 62 

13. 20 

4. 67 

13. 18 

4. 73 

15 

14. 18 

4. 88 | 

14. 16 

4. 95 

14. 14 

5. 01 

14. 12 

5. 07 

1G 

15. 13 

5. 21 

15. 11 

5. 28 

15. 08 

5. 34 

15. 06 

5. 41 

17 

16. 07 

5. 53 

16. 05 

5. 60 

16. 02 

5. 67 

16. 00 

5. 74 

18 

17. 02 

5. 86 

16. 99 

5. 93 

16. 97 

6. 01 

16. 94 

6. 08 

19 

17. 96 

6. 19 | 

17. 94 

6. 26 

17. 91 

6. 34 

17. 88 

6. 42 

20 

18. 91 

6. 51 

18. 88 

6. 59 

18. 85 

6. 68 

18. 82 

6. 76 

21 

19. 86 

6. 84 

19. 83 

6. 92 

19. 80 

7. 01 

19. 76 

7. 10 

22 

20. 80 

7. 16 

20. 77 

7. 25 

20. 74 

7. 34 

20. 71 

7. 43 

23 

21. 75 

7. 49 

21. 71 

7. 58 

21. 68 

7. 68 

21. 65 

7. 77 

24 

22. 69 

7. 81 

22. 66 

7. 91 

22. 62 

8. 01 

22. 59 

8. 11 

25 

23. 64 

8. 14 

23. 60 

8. 24 

23. 57 

8. 35 1 

23. 53 

8. 45 

2G 

24. 58 

8. 46 

24. 55 

8. 57 

24. 51 

8. 68 

24. 47 

8. 79 

27 

25. 53 

8. 79 

25. 49 

8. 90 

25. 45 

9. 01 

25. 41 

9. 12 

28 

26. 47 

9. 12 ! 

26. 43 

9. 23 

26. 39 

9. 35 

26. 35 

9. 46 

29 

27. 42 

9. 44 

27. 38 

9. 56 

27. 34 

9. 68 

27. 29 

9. 80 

30 

28- 37 

9. 77 

28. 32 

9. 89 

28. 28 

10. 01 

28. 24 

10. 14 

35 

3°. 09 

11. 39 

33. 04 

11. 54 

32. 99 

11. 68 

32. 94 

11. 83 

40 

3'. 32 

13. 02 1 

37. 76 

13. 19 

37. 71 

13. 35 

37. 65 

13. 52 

45 

42. 55 

14. 65 

42. 48 

14. 84 

42. 42 

15. 02 

42. fitj 

15. 21 

50 

47. 28 

16. 28 I 

47. 20 

16. 48 

47. 13 

16. 69 

47. 06 

16. 90 

55 

52. 00 

17. 91 

51. 92 

18. 13 

51. 85 

18. 36 

51. 76 

18. 59 

GO 

56. 73 

19. 53 

56. 65 

19. 78 

56. 56 

20. 03 

56. 47 

20. 27 

65 

61. 46 

21. 16 

61. 37 

21. 43 

61. 27 

21. 70 

61. 18 

21. 96 

70 

66. 19 

22. 79 

66. 09 

23. 08 

65. 98 

23. 37 

65. 88 

23. 65 

75 

70. 91 

24. 42 

70. 81 

24. 73 

70. 70 

25. 04 

70. 59 

25. 34 

80 

75. 64 

26. 05 

75. 53 

26. 38 

75. 41 

26. 70 

75. 29 

27. 03 

85 

80. 37 

27. 67 

80. 25 

28. 02 

80. 12 

28. 37 

80. 00 

28. 72 

90 

85. 10 

29. 30 

84. 97 

29. 67 

8% 84 

30. 04 

84. 71 

30. 41 

95 

89. 82 

30. 93 

89. G9 

31. 32 

89. 55 

31. 71 

89. 41 

32. 10 

100 

94. 55 

32. 56 I 

94. 41 

32. 97 

94. 26 

33. 38 

94. 12 

33. 79 

© 

o 

a 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

to 

5 

71 Deg, 

70% Deg. 

70% Deg. 

70% Deg. 



























































































TRAVERSE TABLE. 91 


0> 

V 

c 

C3 

20 Deo. 

20% Deg. 

20% Deg. 

20% Deg. 

CO 

5 

Lat. 

Dcp. 

Lat. 

Dcp. 

Lat. 

Dcp. 

L;t. 

Dcp 

1 

0. 94 

0. 34 

0. 94 

0. 35 

0. 94 

0. 35 

0. 94 

0. 35 

2 

1. 88 

0. 68 

1. 88 

0. 69 

1. 87 

0. 70 

1. 87 

0. 71 

3 

2. 82 

1. 03 

2. 81 

1. 04 

2. 81 

1. 05 

2. 81 

1. 06 ! 

4 

3. 76 

1. 37 

3. 75 

1. 38 

3. 75 

1. 40 

3. 74 

1. 42 

5 

4. 70 

1. 71 

4. 69 

1. 73 

4. 68 

1. 75 

4. 68 

1. 77 

C 

5. 64 

2. 05 

5. 63 

2. 08 

5. 62 

2. 10 

5. G1 

2. 13 j 

7 

6. 58 

2. 39 

6. 57 

2. 42 

6. 56 

2. 45 

6. 55 

2. 48 

8 

7. 52 

2. 74 

7. 51 

2. 77 

7. 49 

2. 80 

7. 48 

2. 83 

9 

8. 46 

3. 08 

8. 44 

3. 12 

8. 43 

3. 15 

8. 42 

3. 19 

10 

9. 40 

3. 42 

9. 38 

3. 46 

9. 37 

3. 50 

9. 35 

3. 54 

11 

10. 34 

3. 76 

10. 32 

3. 81 

10. 30 

3. 85 

10. 29 

3. 90 

12 

11. 28 

4. 10 

11. 26 

4. 15 

11. 24 

4. 20 

11. 22 

4. 25 

13 

12. 22 

4. 45 

12. 20 

4. 50 

12. 18 

4. 55 

12. 16 

4. 61 

14 

13. 16 

4. 79 

13. 13 

4. 85 

13. 11 

4. 90 

13. 09 

4. 96 

15 

14. 10 

5. 13 

14. 07 

5. 19 

14. 05 

5. 25 

14. 03 

5. 31 

16 

15. 04 

5. 47 

15. 01 

5. 54 

14. 99 

5. 60 

14. 96 

5. 67 

17 

15. 97 

5. 81 

15. 95 

5. 88 

15. 92 

5. 95 

15. 90 

6. 02 

18 

16. 91 

6. 16 

16. 89 

6. 23 

16. 86 

6. SO 

16. 83 

6. 38 

19 

17. 85 

6. 50 

17. 83 

6. 58 

17. 80 

6. 65 

17. 77 

6. 73 

20 

18. 79 

6. 84 

18. 76 

6. 92 

18. 73 

7. 00 

18. 70 

7. 09 

21 

19. 73 

7. 18 

19. 70 

7. 27 

19. 67 

7. 35 

19. 64 

7. 44 

22 

20. 67 

7. 52 

20. 64 

7. 61 

20. 61 

7. 70 

20. 57 

7. 79 

23 

21. 01 

7. 87 

21. 58 

7. 96 

21. 54 

8. 05 

21. 51 

8. 15 

24 

22. 55 

8. 21 

22. 52 

8. 31 

22. 48 

8. 40 

22. 44 

8. 50 

25 

23. 49 

8. 55 

23. 45 

8. 65 

23. 42 

8. 76 

23. 38 

8. 86 

26 

24. 43 

8. 89 

24. 39 

9. 00 

24. 35 

9. 11 

24. 31 

9. 21 

27 

25. 37 

9. 23 

25. 33 

9. 35 

25. 29 

9. 46 

25. 25 

9. 57 

28 

26. 31 

9. 58 

26. 27 

9. 69 

20. 23 

9. 81 

26. 18 

9. 92 

29 

27. 25 

9. 92 

27. 21 

10. 04 

27. 16 

10. 16 

27. 12 

10. 27 

30 

28- 19 

10. 26 

28. 15 

10. 38 

28. 10 

10. 51 

28. 05 

10. 63 

35 

32. 89 

11. 97 

32. 84 

12. 11 

32. 78 

12. 20 

32. 73 

12. 40 

40 

37. 59 

13. 68 

37. 53 

13. 84 

37. 47 

14. 01 

37. 41 

14. 17 

45 

42. 29 

15. 39 

42. 22 

15. 58 

42. 15 

15. 76 

42. 08 

15. 94 

50 

46. 98 

17. 10 

46. 91 

17. 31 

46. 83 

17. 51 

46. 76 

17. 71 

55 

51. 68 

18. 81 

51. 60 

19. 04 

51. 52 

19. 26 

51. 43 

19. 49 

60 

56. 38 

20. 52 

56. 29 

20. 77 

56. 20 

21. 01 

56. 11 

'21. 26 

65 

61. 08 

22. 23 

60. 98 

22. 50 

60. 88 

22. 76 

60. 78 

23. 03 

70 

65. 78 

23. 94 

65. 67 

24. 23 

65. 57 

24. 51 

65. 46 

24. 80 

75 

70. 48 

25. 65 

70. 36 

25. 96 

70. 25 

26. 27 

70. 14 

26. 57 

80 

75. 18 

27. 36 

75. 06 

27. 69 

74. 93 

28. 02 

74. 81 

28. 34 

85 

79. 87 

29. 07 

79- 75 

29. 42 

79. 62 

29. 77 

79. 49 

30. 11 

90 

84. 57 

30. 78 

84- 44 

31. 15 

84. 30 

31. 52 

84. 16 

31. 89 

95 

89. 27 

32. 49 

89. 13 

32. 88 

88. 98 ! 

33. 27 

88. 84 

33. GG 

100 

93. 97 

34. 20 

93- 82 

34. 61 

93. 67 | 

35. 02 

93. 51 

35. 43 

oJ 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. j 

Lat. 

Dep. 

Lat. 

53 

+-> 

*/3 

A 

70 Deg. 

09% Deg. 

69% Deg. 

69% Deg. 


















































































































92 TRAVERSE TABLE. 


o 

c 

CS 

21 Deo. 

21 H 

Deg 


21X 

Deg. 

21 % 

Deg. 

•4 

P 

Lat. 

Th 

‘P- 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

Dep. - 

1 

0 . 

93 

0 . 

36 

0 . 

93 

0 . 

36 

0. 93 

0. 37 

0 . 

93 

0. 37 

2 

1. 

87 

0 . 

72 

1. 

86 

0 . 

72 

1. 86 

0. 73 

1. 

86 

0. 74 

3 

2. 

80 

1. 

08 

2. 

80 

1 . 

09 

2. 79 

1. 10 

2. 

79 

1. 11 

4 

3. 

73 

1. 

43 

3. 

73 

1 . 

45 

3. 72 

1. 47 

3. 

72 

1. 48 

5 

4. 

67 

1. 

79 

4. 

66 

1 . 

81 

4. 65 

1. 83 

4. 

64 

1. 85 

G 

5. 

60 

2. 

15 

5. 

59 

2. 

17 

5. 58 

2. 20 

5. 

57 

2. 22 

7 

6. 

54 

2. 

51 

6. 

52 

2. 

54 

6. 51 

2. 57 

6. 

50 

2. 59 

8 

7. 

47 

2. 

87 

7. 

46 

2. 

90 

7. 44 

2. 93 

7. 

43 

2. 96 

9 

8. 

40 

3. 

23 

8. 

39 

3. 

26 

8. 37 

3. 30 

8. 

36 

3. 34 

10 

9. 

34 

3. 

58 

9. 

32 

3. 

62 

9. 30 

3. 67 

9. 

29 

3. 71 

11 

10. 

27 

3. 

94 

10. 

25 

3. 

99 

10. 23 

4. 03 

10. 

22 

4. 08 

12 

11. 

20 

4. 

30 

11. 

18 

4. 

35 

11. 17 

4. 40 

11. 

15 

4. 45 

13 

12. 

14 

4. 

66 

12. 

12 

4. 

71 

12. 10 

4. 76 

12. 

07 

4. 82 

14 

13. 

07 

5. 

02 

13. 

05 

5. 

07 

13. 03 

5. 13 

13. 

00 

5. 19 

15 

14. 

00 

5. 

38 

13. 

98 

5. 

44 

13. 96 

5. 50 

13. 

93 

5. 56 

16 

14. 

94 

5. 

73 

14. 

91 

5. 

80 

14. 89 

5. 86 

14. 

86 

5. 93 

17 

15. 

87 

6. 

09 

15. 

84 

6 . 

16 

15. 82 

6. 23 

15. 

79 

6. 30 

18 

16. 

80 

6. 

45 

16. 

78 

6. 

52 

16. 75 

6. 60 

16. 

72 

6. 67 

19 

17. 

74 

6. 

81 

17. 

71 

6. 

89 

17. 68 

6. 96 

17. 

65 

7. 04 

20 

18. 

67 

7. 

17 

18. 

64 

7. 

25 

18. 61 

7. 33 

18. 

58 

7. 41 

21 

19. 

61 

7. 

53 

19. 

57 

7. 

61 

19. 54 

7. 70 

19. 

50 

7. 78 

22 

20. 

54 

7. 

88 

20. 

50 

7. 

97 

20. 47 

8. 06 

20. 

43 

8. 15 

23 

21. 

47 

8. 

24 

21. 

44 

8. 

34 

21. 40 

8. 43 

21. 

36 

8. 52 

24 

22. 

41 

8. 

60 

22. 

37 

8. 

70 

22. 33 

8. 80 

22. 

29 

8. 89 

25 

23. 

34 

8. 

96 

23. 

30 

9. 

06 

23. 26 

9. 16 

23. 

22 

9. 26 

26 

24. 

27 

9. 

32 

24. 

23 

9. 

42 

24. 19 

9. 53 

24. 

15 

9. 63 

27 

25. 

21 

9. 

68 

25. 

16 

9. 

79 

25. 12 

9. 90 

25. 

08 

10. 01 

28 

26. 

14 

10. 

03 

26. 

10 

10. 

15 

26. 05 

10. 26 

26. 

01 

10. 38 

29 

27. 

07 

10. 

39 

27. 

03 

10. 

51 

26. 98 

10. 63 

26. 

94 

10. 75 

30 

28. 

01 

10. 

75 

27. 

96 

10. 

87 

27. 91 

11. 00 

27. 

86 

11. 12 

35 

32. 

68 

12. 

54 

32. 

62 

12. 

69 

32. 56 

12. 83 

32. 

51 

12. 97 

40 

37. 

34 

14. 

33 

37. 

28 

14. 

50 

37. 22 

14. 66 

37. 

15 

14. 82 

45 

42. 

01 

16. 

13 

41. 

94 

16. 

31 

41. 87 

16. 49 

41. 

80 

16. 68 

50 

46. 

68 

17. 

92 

46. 

60 

18. 

12 

46. 52 

18. 33 

46. 

44 

18. 53 

55 

51. 

35 

19. 

71 

51. 

26 

19. 

93 

51. 17 

20 16 

51. 

08 

20. 38 

60 

56. 

01 

21. 

50 

55. 

92 

21. 

75 

55. 83 

21. 99 

55. 

73 

22. 23 

65 

60. 

68 

23. 

29 

60. 

58 

23. 

56 

60. 48 

23. 82 

60. 

37 

24. 09 

70 

65. 

35 

25. 

09 

65. 

24 

25. 

37 

65. 13 

25. 66 

65. 

02 

25. 94 

75 

70. 

02 

26. 

88 

69. 

90 

27. 

18 

69. 78 

27. 49 

69. 

66 

27. 79 

80 

74. 

69 

28. 

67 

74. 

56 

29. 

00 

74. 43 

29. 32 

74. 

30 

29. 64 

85 

79. 

35 

30. 

46 

79. 

22 

30. 

81 

79. 09 

31. 15 

78. 

95 

31. 50 

90 

84. 

02 

32. 

25 

83. 

88 

32. 

62 

83. 74 

32. 99 

83. 

59 

33. 35 

95 

88. 

69 

34. 

04 

88. 

54 

34. 

43 

88. 39 

34. 82 

88. 

24 

35. 20 

100 

93. 

36 

35. 

84 ; 

93. 

20 

36. 

24 

93. 04 

36. 65 

92. 

88 

37. 06 

V 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

m 

p 

69 Deg. 

63 % 

Deg 


68)^ Deg. 

68^ Deg. 





































































TRAVERSE 

TABLE. 





93 

Distance. 

22 Deo. 

22% Deg. 

22% Deg. 

22% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

1 

0. 93 

0. 

37 

0. 

93 

0. 38 

0. 92 

0. 

38 

0. 

92 

0. 

39 

2 

1. 85 

0. 

75 

1. 

85 

0. 76 

1. 85 

0. 

77 

1. 

84 

0. 

77 

3 

2. 78 

1. 

12 

2. 

78 

1. 14 

2. 77 

1. 

15 

2. 

77 

1. 

18 

4 

3. 71 

1. 

50 

3. 

70 

1. 51 

3. 70 

1. 

53 

3. 

69 

1. 

55 

5 

4. 64 

1. 

87 

4. 

63 

1. 89 

4. 62 

1. 

91 

4. 

61 

1. 

93 

C 

5. 56 

2. 

25 

5. 

55 

2. 27 

5. 54 

2. 

30 

5. 

53 

2. 


7 

6. 49 

2. 

62 

6. 

43 

2. 65 

6. 47 

2. 

68 

6. 

46 

2. 

71 

8 

7. 42 

3. 

00 

7. 

40 

3. 03 

7. 39 

3. 

06 

7. 

38 

3. 

09 

9 

8. 34 

3. 

87 

8. 

33 

3. 41 

8. 81 

3. 

44 

8. 

80 

3. 

48 

10 

9. 27 

3. 

75 

9. 

26 

3. 79 

9. 24 

3. 

83 

9. 

22 

3. 

87 

11 

10. 20 

4. 

12 

10. 

18 

4. 17 

10. 16 

4. 

21 

10. 

14 

4. 

25 

12 

11. 13 

4. 

50 

11. 

11 

4. 54 

11. 09 

4. 

59 

11. 

07 

4. 

64 

13 

12. 05 

4. 

87 

12. 

03 

4. 92 

12. 01 

4. 

97 

11. 

99 

5. 

03 

14 

12. 98 

5. 

24 

12. 

96 

5. 30 

12. 93 

5. 

36 

12. 

91 

5. 

41 

15 

13. 91 

5. 

62 

13. 

88 

5. 68 

13. 86 

5. 

74 

13. 

83 

5. 

80 

10 

14. 83 

5. 

99 

14. 

81 

6. 06 

14. 78 

6. 

12 

14. 

76 

6. 

19 

17 

15. 76 

6 . 

37 

15. 

73 

6. 44 

15. 71 

6 . 

51 

15. 

68 

6. 

57 

18 

16. 69 

6. 

74 

16. 

66 

6. 82 

16. 63 

6. 

89 

16. 

60 

6. 

96 

19 

17. 62 

7. 

12 

17. 

59 

7. 19 

17. 55 

7. 

27 

17. 

52 

7. 

35 

20 

18. 54 

7. 

49 

18. 

51 

7. 57 

18. 48 

7. 

65 

18. 

44 

7. 

73 

21 

19. 47 

7. 

87 

19. 

44 

7. 95 

19. 40 

8. 

04 

19. 

37 

8. 

12 

22 

20. 40 

8. 

24 

20. 

36 

8. 33 

20. 33 

8. 

42 

20. 

29 

8. 

51 

23 

21. 33 

8. 

62 

21. 

29 

8. 71 

21. 25 

8. 

80 

21. 

21 

8. 

89 

24 

22. 25 

8. 

99 

22. 

21 

9. 09 

22. 17 

9. 

18 

22. 

13 

9. 

28 

25 

23. 13 

9. 

37 

23. 

14 

9. 47 

23. 10 

9. 

57 

23. 

05 

9. 

67 

2G 

24. 11 

9. 

74 

24. 

03 

9. 84 

24. 02 

9. 

95 

23. 

98 

10. 

05 

27 

25. 03 

10. 

11 

21. 

99 

10. 22 

24. 94 

10. 

83 

24. 

90 

10. 

44 

28 

25. 96 

10. 

49 

25. 

92 

10. 60 

25. 87 

10. 

72 

25. 

82 

10. 

88 

29 

26. 89 

10. 

86 

26. 

84 

10. 98 

26. 79 

11. 

10 

26. 

74 

11. 

21 

30 

27. 82 

11. 

24 

27.. 

77 

11. 36 

27. 72 

11. 

48 

27. 

67 

11. 

60 

35 

32. 45 

13. 

11 

82. 

89 

13. 25 

32. 34 

13. 

39 

32. 

28 

13. 

58 

40 

37. 09 

14. 

93 

37. 

02 

15. 15 

86. 96 

15. 

81 

86. 

89 

15. 

47 j 

45 

41. 72 

16. 

86 

41. 

65 

17. 04 

41. 57 

17. 

22 

41. 

50 

17. 

40 [j 

50 

46. 36 

18. 

73 

46. 

28 

18. 93 

46. 19 

19. 

13 

46. 

11 

19. 

84 j 

55 

51. 00 

20. 

60 

50. 

90 

20. 83 

50. 81 

21. 

05 

50. 

72 

21. 

27 

60 

55. 63 

22. 

48 

55. 

53 

22. 72 

55. 43 

22. 

96 

55. 

33 

23. 

20 

65 

60. 27 

24. 

35 

60. 

16 

24. 61 

60. 05 

24. 

87 

59. 

94 

25. 

14 

70 

64. 90 

26. 

22 

64. 

79 

26. 51 

64. 67 

26. 

79 

64. 

55 

27. 

07 j< 

75 

69. 54 

28. 

10 

69. 

42 

28. 40 

69. 29 

28. 

70 

69. 

17 

29. 

00 g 

80 

74. 17 

29. 

97 

74. 

04 

80. 29 

73. 91 

30. 

61 

73. 

78 

30. 

94 i 

85 

78. 81 

31. 

84 

78. 

67 

32. 19 

78. 53 

32. 

53 

78. 

39 

32. 

87 

90 

83. 45 

33. 

71 

83. 

30 

34. 03 

83. 15 

84. 

44 

83. 

00 

34. 

80 

95 

88. 03 

35. 

59 

87. 

93 

35. 97 

87. 77 

86. 

35 

87. 

61 

36. 

74 

too 

92. 72 

37. 

43 

92. 

55 

37. 83 

12. 89 

88. 

27 

92. 

22 

88. 

67 

c 

P 

/. 

c ! 

Dep. 

Lat. 

Dep. 

L ft. 

Dep-.-' 

Lat. 

Dep. 

Lat. 

63 Dig. 

C 

7% Deg. 

G7% Dlg. 

* 

G7% Di.g 




































































































94 TRAVERSE TABLE. 


6 

5 

23 Deg. 



Deg 

• 

23X 

Deg. 


23% Deg. 

to 

Lat. 

Dep. 

Lat. 

Dep. 

lift!* 

Dep. 

Lat. 

Dep. 

1 

0. 

92 

0. 

39 

0. 

92 

0. 

39 

0. 

92 

0. 

40 

0. 92 

0. 

40 

2 

1. 

84 

0. 

78 

1. 

84 

0. 

79. 

l. 

83 

0. 

80 

1. 83 

0. 

81 

3 

2. 

76 

1. 

17 

2. 

76 

1. 

18 

2. 

75 

1. 

20 

2. 75 

1. 

21 

4 

3. 

63 

1. 

56 

3. 

68 

1. 

58 

3. 

67 

1. 

59 

3. 66 

1. 

61 

5 

4. 

60 

1. 

95 

4. 

59 

1. 

97 

4. 

59 

1. 

99 

4. 58 

2. 

01 

6 

5. 

52 

2. 

34 

5. 

51 

2. 

37 

5. 

50 

2. 

39 

5. 49 

2. 

42 

7 

6. 

44 

2. 

74 

6. 

43 

2. 

76 

6. 

42 

2. 

79 

6. 41 

2. 

82 

8 

7. 

36 

3. 

13 

7. 

35 

3. 

16 

7. 

34 

3. 

19 

7. 32 

3. 

22 

9 

8. 

23 

3. 

52 

8. 

27 

3. 

55 

8. 

25 

3. 

59 

8. 24 

3. 

62 

10 

9. 

20 

3. 

91 

9. 

19 

3. 

95 

9. 

17 

3. 

99 

9. 15 

4. 

03 

11 

10. 

13 

4. 

30 

10. 

11 

4. 

34 

10. 

09 

4. 

39 

10. 07 

4. 

43 

12 

11. 

05 

4. 

69 

11. 

03 

4. 

74 

1 11. 

00 

4. 

78 

j 10. 98 

4. 

83 

13 

11. 

97 

5. 

03 

11. 

94 

5. 

13 

1 

92 

5. 

18 

11. 90 

5. 

24 

14 

12. 

89 

5. 

47 

12. 

86 

5. 

53 

12. 

84 

5. 

53 

12. 81 

5. 

64 

15 

13. 

81 

5. 

86 

13. 

78 

5. 

92 1 

13. 

76 

5. 

98 

13. 73 

6. 

04 

16 

14. 

73 

6. 

25 I 

14. 

70 

6. 

32 

14. 

67 

6. 

38 

14. 64 

6. 

44 

17 

15. 

65 

6. 

64 

15. 

62 

6. 

71 

15. 

59 

6. 

78 

15. 56 

6. 

85 

18 

16. 

57 

7. 

03 

16. 

54 

7. 

11 

16. 

51 

7. 

13 

16. 48 

7. 

25 

19 

17. 

49 

7. 

42 

17. 

46 

7. 

50 

17. 

42 

7. 

53 

17. 39 

7. 

65 

20 

18. 

41 

7. 

81 

18. 

38 

7. 

89 

18. 

34 

7. 

97 

18. 31 

8. 

05 

21 

19. 

33 

8. 

21 

19. 

29 

8. 

29 

19. 

26 

8. 

37 

19. 22 

8. 

46 

22 

20. 

25 

8. 

60 

20. 

21 

8. 

68 

20. 

18 

8. 

77 

20. 14 

8. 

86 

23 

21. 

17 

8. 

99 

21. 

13 

9. 

03 

21. 

09 

9. 

17 

21. 05 

9. 

26 

24 

22. 

09 

9. 

33 

22. 

05 

9. 

47 

22. 

01 

9. 

57 

21. 97 

9. 

67 

25 

23. 

01 

9. 

77 

22. 

97 

9. 

87 

22. 

93 

9. 

97 

22. 88 

10. 

07 

26 

23. 

93 

10. 

16 

23. 

89 

10. 

26 

23. 

84 

10. 

37 1 

23. 80 

10. 

47 

27 

24. 

85 

10. 

55 

24. 

81 

10. 

66 

24. 

76 

10. 

77 | 

24. 71 

10. 

87 

28 

23. 

77 

19. 

94 

25. 

73 

11. 

05 

25. 

68 

11. 

16 j 

25. 63 

11. 

23 

29 

26. 

69 

11. 

33 

26. 

64 

11. 

45 

26. 

59 

11. 

56 ; 

26. 54 

11. 

68 

30 

27. 

62 

11. 

72 

27. 

56 

11. 

84 

! 27. 

51 

11. 

96 ; 

27. 46 

12. 

08 

33 

32. 

22 

13. 

63 

32. 

16 

13. 

82 

32. 

10 

13. 

96 

32. 04 

14. 

10 

40 

36. 

82 

15. 

63 

36. 

75 

15. 

79 

36. 

63 

15. 

95 | 

36. 61 

16. 

11 

45 

41. 

42 

17. 

53 

41. 

35 

17. 

76 

41. 

27 

17. 

94 i 

41. 19 

18. 

12 

50 

46. 

03 

19. 

54 

45. 

94 

19. 

74 

45. 

85 

19. 

94 j 

45. 77 

20. 

14 

55 

50. 

63 

21. 

49 

50. 

53 

21. 

71 

50. 

44 

21. 

93 

50. 34 

22. 

15 

60 

55. 

23 

23. 

44 

5o. 

13 

23. 

68 

55. 

03 

23. 

92 

54. 92 

24. 

16 

65 

59. 

83 

2o. 

40 

59. 

72 

25. 

66 

59. 

61 

25. 

92 

59. 50 

26. 

13 

70 

64. 

44 

27. 

35 

64. 

32 

27. 

63 

64. 

19 

27. 

91 

64. 07 

28. 

19 

75 

69. 

04 

29. 

30 

63. 

91 

29. 

61 

63. 

78 

29. 

91 

68. 65 

30. 

21 

80 

73. 

64 

81. 

26 

73. 

50 

31. 

53 

73. 

33 

31. 

90 

73. 22 

32. 

22 

85 

78. 

24 

33. 

21 

73. 

19 

33. 

55 

77. 

95 

33. 

89 

77. 80 

34. 

23 

90 

82. 

85 

85. 

17 

82. 

69 

35. 

53 

82. 

54 

35. 

89 

82. 33 

3,6. 

25 

95 

87. 

45 

37. 

12 

87. 

29 

37. 

50 

87. 

13 

37. 

83 

86. 95 

38. 

26 

100 

92. 

05 

39. 

07 

91. 

88 

39. 

47 

91. 

71 

39. 

87 

91. 53 

40. 

27 

*5 

o 

fl 

Dep. 

L: 

it. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

cl 

aft 

...A- ■ 


G7 Deg. 




Deg 


j 1 

• I 

° ll 

Deg 


G6% 

Deg 






























































































TRAVERSE 

TABLE. 


95 

o 

V 

G 

Cj 

•*-> 

24 Deo. 

24 % Di o. 

24 .% Deo. 

24 % Deg. 

E ® 

Lat. 

Di'p. 

Lat. 

Dep. 

Lat. 

Dep. 

I Lat. 

Dep. 

1 

0. 91 

0. 41 

0. 91 

0. 41 

0. 91 

0. 41 

0. 91 

0. 42 

2 

1. 83 

0. 81 

1. 82 

0. 82 

1. 82 

0. 83 

1. 82 

0. 84 

3 

2. 74 

1. 22 

2, 74 

1. 23 

2. 73 

1. 24 

| 2. 72 

1. 26 

4 

3. 65 

1. G3 

3. 65 

1. 64 

3. 64 

1. 66 

3. 63 

1. 67 

5 

4. 57 

2. 03 

4. 56 

2. 05 

4. 55 

2. 07 

4. 54 

2. 09 

Jj 6 

5. 48 

2. 44 

5. 47 

2. 46 

5. 46 

2. 49 

5. 45 

2. 51 

7 

6. 39 

2. 85 

G. 88 

2. 87 

6. 87 

2. 00 

6. 36 

2. 93 

8 

7. 31 

3. 25 

7. 29 

3. 29 

7. 28 

8. 32 

7. 27 

3. 35 

9 

8. 22 

3. 66 

8. 21 

3. 70 

8. 19 

3. 73 

8. 17 

3. 77 

10 

9. 11 

4. 07 

9. 12 

4. 11 

9. 10 

4. 15 

9. 08 

4. 19 

11 

10. 05 

4. 47 

10. 03 

4. 52 

10. 01 

4. 56 

9. 09 

4. 61 

12 

10. 96 

4. 88 

10. 94 

4. 93 

10. 92 

4. 98 

10. 90 

5. 02 

13 

11. 88 

5. 29 

11. 85 

5. 34 

11. 83 

5. 39 

; 11. 81 

5. 44 

14 

12. 79 

5. 69 

12. 76 

5. 75 

12. 74 

5. 81 

1 12. 71 

5. 86 

15 

13. 70 

6. 10 

| 13. 68 

6. 16 

13. 65 

6. 22 

13. 62 

6. 28 

16 

14. 62 

6. 51 

14. 59 

6. 57 

14. 56 

G. 64 

' 14. 53 

6. 70 

17 

15. 53 

6. 92 

15. 50 

6. 98 

15. 47 

7. 05 

1 15. 44 

7. 12 

18 

16. 44 

7. 32 

16. 41 

7. 39 

16. 38 

7. 46 

16. 35 

7. 54 

19 

17. 36 

7. 73 

17. 32 

7. 80 

17. 29 

7. 83 

17. 25 

7. 05 

20 

18. 27 

8. 13 

i 18. 24 

8. 21 

18. 20 

8. 29 

18. 16 

8. 37 

21 

19. 18 

8. 54 

19. 15 

8. 63 

19. 11 

8.-71 

19. 07 

8. 79 

22 

20. 10 

8. 95 

20. 06 

9. 04 

20. 02 

9. 12 

19. 98 

9. 21 

23 

21. 01 

9. 35 

1 20. 97 

9. 45 

20. 93 

9. 54 

20. 89 

9. 63 

24 

21. 93 

9. 76 

1 21. 88 

9. 86 

21. 84 

9. 95 

21. 80 

10. 05 

25 

22. 84 

10. 17 

| 22. 79 

10. 27 

22. 75 

10. 37 

22. 70 

10. 47 

26 

23r 75 

10. 58 

23. 71 

10. 68 

23. 66 

10. 78 

23. 61 

10. 89 

27 

24. 67 

10. 98 

24. 62 

11. 09 

24. 57 

11. 20 

24. 52 

11. 30 

28 

25. 58 

11. 39 

! 25. 53 

11. 50 

25. 48 

11. 61 

25. 43 

11. 72 

2J 

26. 49 

11. 80 

26. 44 

11. 91 

26. 39 

12. 03 

i 26. 34 

12. 14 

30 

33 

27. 41 

12. 20 

27. 35 

12. 32 

27. 30 

12. 44 

27. 24 

12. 56 

31. 97 

14. 24 

31. 91 

14. 38 

31. 85 

14. 51 

31. 78 

14. 65 

40 

36. 51 

16. 27 

36. 47 

16. 43 

36. 40 

16. 59 

! 36. 33 

16. 75 

45 

41. 11 

18. 30 

41. 03 

18. 48 

40. 95 

18. 66 

40. 87 

18. 84 

50 

45. 63 

20. 34 

45. 59 

20. 54 

45. 50 

20. 73 

45. 41 

20. 93 

55 

50. 24 

22. 37 

50. 15 

22. 59 

50. 05 

22. 81 

49. 95 

23. 03 

60 

54. 81 

24. 40 

54. 71 

24. 64 

54. 60 

24. 88 

54. 49 

25. 12 

65 

59. 33 

26. 44 

59. 26 

26. 70 

59. 15 

26. 96 

59. 03 

27. 21 

1 70 

63. 95 

28. 47 

63. 82 

28. 75 

63. 70 

29. 03 

63. 57 

29. 31 

75 

68. 52 

30. 51 

68. 38 

30. 80 

68. 25 

31. 10 

68. 11 

31. 40 

80 

73. 08 

32. 54 

72. 94 

32. 86 

72. 80 

33. 18 

72. 65 

33. 49 

85 

77. 65 

34. 57 

77. 50 

34. 91 

77. 35 

35. 25 

77. 19 

35. 59 

90 

82. 22 

36. 61 

82. 06 

36. 96 

81. 90 

37. 32 

81. 73 

37. 68 

95 

86. 79 

33. 64 

86. 62 

39. 02 

86. 45 

39. 40 

86. 27 

39. 77 

100 

91. 35 

40. 67 

91. 18 

41. 07 

01. 00 

41. 47 

90. 81 

41. 87 

6 

o 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

-*-> 
c n 

J5. 

66 Deo. 

65% Deo. 

6Deg. 

65 % Deg. 
























































































96 



TRAVERSE 

TABLE. 



Distance. 

25 Deg. 

25% 

Deg. 

25% Deg. 

25% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 91 

0. 42 

0. 90 

0. 43 

0. 90 

0. 43 

0. 90 

0. 43 

2 

1. 81 

0. 85 

1. 81 

0. 85 

1. 81 

0. 86 

1. 80 

0. 87 

3 

2. 72 

1. 27 

2. 71 

1. 28 

2. 71 

1. 29 

2. 70 

1. 30 

j 4 

3. 63 

1. 69 

3. 62 

1. 71 

3. 61 

1. 72 

3. 60 

1. 74 

5 

4. 53 

2. 11 

4. 52 

2. 13 

4. 51 

2. 15 

4. 50 

2. 17 

6 

5. 44 

2. 54 

5. 43 

2. 56 

5. 42 

2. 58 

5. 40 

2. 61 

7 

6. 34 

2. 96 

6. 33 

2. 99 

6. 32 

3. 01 

6. 30 

3. 04 

8 

7. 25 

3. 38 

7. 24 

3. 41 

7. 22 

3. 44 

7. 21 

3. 48 

9 

8. 16 

3. 80 

8. 14 

3. 84 

8. 12 

3. 87 

8. 11 

3. 91 

10 

9. 06 

4. 23 

9. 04 

4. 27 

9. 03 

4. 31 

9. 01 

4. 34 

11 

9. 97 

4. 65 

9. 95 

4. 60 

9. 93 

4. 74 

9. 91 

4. 78 

12 

10. 88 

5. 07 

10. 85 

5. 12 

10. 83 

5. 17 

10. 81 

5. 21 

13 

11. 78 

5. 49 

11. 76 

5. 55 

11. 73 

5. 60 

11. 71 

5. 65 

14 

12. 69 

5. 92 

12. 66 

5. 97 

12. 64 

6. 03 

12. 61 

6. 08 

15 

13. 59 

6. 34 

13. 57 

6. 40 

13. 54 

6. 46 

13. 51 

6. 52 

16 

14. 50 

6. 76 

14. 47 

6. 83 

14. 44 

6. 89 

14. 41 

6. 95 

17 

15. 41 

7. 18 

15. 38 

7. 25 

15. 34 

7. 32 

15. 31 

7. 39 

18 

16. 31 

7. 61 

16. 28 

7. 63 

16. 25 

7. 75 

16. 21 

7. 82 

19 

17. 22 

8. 03 

17. 13 

8. 10 

17. 15 

8. 18 

17. 11 

8. 25 

20 

18. 13 

8. 45 

18. 09 

8. 53 

18. 05 

8. 61 

18. 01 

8. 69 

21 

19. 03 

'8. 87 

18. 99 

8. 96 

18. 95 

9. 04 

18. 91 

9. 12 

22 

19. 94 

9. 30 

19. 90 

9. 38 

19. 86 

9. 47 

19. 82 

9. 56 

23 

20. 85 

9. 72 

20. 80 

9. 81 

20. 76 

9. 90 

20. 72 

9. 99 

24 

21. 75 

10. 14 

21. 71 

10. 24 

21. 66 

10. 33 

21. 62 

10. 43 

25 

22. 66 

10. 57 

22. 61 

10. 66 

22. 56 

10. 76 

22. 52 

10. 86 

26 

23. 56 

10. 99 

23. 52 

11. 09 

23. 47 

11. 19 

23. 42 

11. 30 

27 

24. 47 

11. 41 

24. 42 

11. 52 

24. 37 

11. 62 

24. 32 

11. 73 

28 

25. 33 

11. 83 

25. 32 

11. 94 

25. 27 

12. 05 

25. 22 

12. 16 

29 

26. 28 

12. 26 

26. 23 

12. 37 

26. 17 

12. 48 

26. 12 

12. 60 

30 

27. 19 

12. 63 

27. 13 

12. 80 

27. 08 

12. 92 

27. 02 

13. 03 

35 

31. 72 

14. 79 

31. 66 

14. 93 

31. 59 

15. 07 

31. 52 

15. 21 

40 

36. 25 

16. 90 

36. 18 

17. 06 

36. 10 

17. 22 

36. 03 

17. 38 

45 

40. 78 

19. 02 

40. 70 

19. 20 

40. 62 

19. 37 

40. 53 

19. 55 

50 

45. 32 

21. 13 

45. 22 

21. 33 

45. 13 

21. 53 

45. 03 

21. 72 

55 

49. 85 

23. 24 

49. 74 

23. 46 

49. 64 

23. 68 

49. 54 

23. 89 

60 

54. 38 

25. 36 

54. 27 

25. 59 

54. 16 

25. 83 

54. 04 

26. 07 

65 

58. 91 

27. 47 

58. 79 

27. 73 

58. 67 

27. 98 

58. 55 

28. 24 

70 

63. 44 

29. 58 

63. 31 

29. 86 

63. 18 

30. 14 

63. 05 

30. 41 

75 

67. 97 

31. 70 

67. 83 

31. 99 

67. 69 

32. 29 

67. 55 

32. 58 

80 

72. 50 

33. 81 

72. 36 

34. 13 

72. 21 

34. 44 

72. 06 

34. 76 

85 

77. 04 

35. 92 

76. 88 

36. 26 

76. 72 

36. 59 

76. 56 

36. 93 

90 

81. 57 

38. 04 

81. 40 

38. 39 

81. 23 

38. 75 

81. 06 

39. 10 

95 

86. 10 

40. 15 

85. 92 

40. 52 

85. 75 

40. 90 

85. 57 

41. 27 

100 

GO. 63 

43. 26 

90. 45 

42. 66 

90. 26 

43. Oo 

90. 07 

, 43. 44 

Distance. 

i 

Dep. 

Lat. 

Dep. 

Latr — 

- Dep:"- 

Latr. — 

Depr. ' 

- Lat. 

65 Deg. 

64% Deg. 

64% Deg. 

64% Deg. 




































































































TRAVERSE 

TABLE. 


97 

I 

3 

+-> 

C0 

5 

26 Deo. 

26% Deg. 

26% Deg. 

a 

26% Deg. 

Lat. 

Dep. 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dcp 

1 

0. 90 

0. 44 

0. GO 

0. 44 

0. 89 

0. 45 

0. 89 

0. 45 

2 

1. 80 

0. 88 

1. 79 

0. 88 

1. 79 

0. 89 

1. 79 

0. 90 

3 

2. 70 

1. 32 

2. 69 

1. 33 

2. 68 

1. 34 

2. 68 

1. 35 

4 

3. 60 

1. 75 

3. 59 

1. 77 

3. 58 

1. 78 

3. 57 

1. 80 

5 

4. 49 

2. 19 

4. 48 

2. 21 

4. 47 

2. 23 

4. 46 

2. 25 

6 

5. 39 

2. 63 

5. 38 

2. 65 

5. 37 

2. 68 

5. 36 

2. 70 | 

7 

6. 29 

3. 07 

6. 28 

3. 10 

6. 26 

3. 12 

6. 25 

3. 15 

8 

7. 19 

3. 51 

7. 17 

3. 54 

7. 16 

3. 57 

7. 14 

3. 60 

9 

8. 09 

3. 95 

8. 07 

3. 98 

8. 05 

4. 02 

8. 04 

4. 05 

10 

8. 99 

4. 38 

8. 97 

4. 42 

8. 95 

4. 46 

8. 93 

4. 50 

11 

9. 89 

4. 82 

9. 87 

4. 87 

9. 84 

4. 91 

9. 82 

4. 95 

12 

10. 79 

5. 26 

10. 76 

5. 31 

10. 74 

5. 35 

10. 72 

5. 40 

13 

11. 68 

5. 70 

11. 66 

5. 75 

11. 63 

5. 80 

11. 61 

5. 85 

14 

12. 58 

6. 14 

12. 56 

6. 19 

12. 53 

6. 25 

12. 50 

6. 30 

15 

13. 48 

6. 58 

13. 45 

6. 63 

13. 42 

6. 69 

13. 89 

6. 75 

16 

14. 38 

7. 01 

14. 35 

7. 08 

14. 32 

7. 14 

14. 29 

7. 20 

17 

15. 28 

7. 45 

15. 25 

7. 52 

15. 21 

7. 59 

15. 18 

7. 65 

18 

16. 18 

7. 89 

16. 14 

7. 96 

16. 11 

8. 03 

16. 07 

8. 10 

19 

17. 08 

8. 33 

17. 04 

8. 40 

17. 00 

8. 48 

16. 97 

8. 55 

20 

17. 98 

8. 77 

17. 94 

8. 85 

17. 90 

8. 92 

17. 86 

9. 00 

21 

18. 87 

9. 21 

18. 83 

9. 29 

18. 79 

9. 37 

18. 75 

9. 45 

22 

19. 77 

9. 64 

19. 73 

9. 73 

19. G9 

9. 82 

19. 65 

9. 90 

23 

20. 67 

10. 08 

20. 63 

10. 17 

20. 58 

10. 26 

20. 54 

10. 35 

24 

21. 57 

10. 52 

21. 52 

10. 61 

21. 48 

10. 71 

21. 43 

10. 80 

25 

22. 47 

10. 96 

22. 42 

11. 06 

22. 37 

11. 15 

22. 32 

11. 25 

26 

23. 37 

11. 40 

23. 32 

11. 50 

23. 27 

11. 60 

23. 22 

11. 70 

27 

24. 27 

11. 84 

24. 22 

11. 94 

24. 16 

12. 05 

24. 11 

12. 15 

28 

25. 17 

12. 27 

25. 11 

12. 38 

25. 06 

12. 49 

25. 00 

12. 60 

29 

26. 06 

12. 71 

26. 01 

12. 83 

25. 95 

12. 94 

25. 90 

13. 05 

30 

26. 96 

13. 15 

26. 91 

13. 27 

26. 85 

13. 39 

26. 79 

13. 50 

35 

31. 46 

15. 34 

31. 39 

15. 48 

31. 32 

15. 62 

31. 25 

15. 75 

40 

35. 95 

17. 53 

35. 87 

17. 69 

35. 80 

17. 85 

35. 72 

18. 00 

45 

40. 45 

19. 73 

40. 36 

19. 90 

40. 27 

20. 08 

40. 18 

20. 25 

50 

44. 94 

21. 92 

44. 84 

22. 11 

44. 75 

22. 31 

44. 65 

22. 50 

55 

49. 43 

24. 11 

49. 33 

24. 33 

49. 22 

24. 54 

49. 11 

24. 76 

60 

53. 93 

26. 80 

53. 81 

26. 54 

53. 70 

26. 77 

53. 58 

27. 01 

65 

58. 42 

28. 49 

58. 30 

28. 75 

58. 17 

29. 00 

58. 04 

29. 26 

70 

62. 92 

30. 69 

62. 78 

30. 96 

62. 65 

31. 23 

62. 51 

31. 51 

75 

67. 41 

32. 88 

67. 27 

33. 17 

67. 12 

33. 46 

66. 97 

33. 76 

80 

71. 90 

35. 07 

71. 75 

35. 38 

71. 59 

35. 70 

71. 44 

36. 01 

85 

76. 40 

37. 26 

76. 23 

37. 59 

76. 07 

37. 93 

75. 90 

38. 26 

90 

80. 89 

39. 45 

80. 72 

39. 81 

80. 54 

40. 16 

80. 37 

40. 51 

95 

85. 39 

41. 65 

85. 20 

42. 02 

85. 02 

42. 39 

84. 83 

42. 76 

100 

89. 88 

43. 84 

89. 69 

44. 23 

89. 49 

44. 62 

89. 30 

45. 01 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

LaT. 

Dep7~ 

Xat. 

64 Deg. 

63% Deo. 

63% Deg. 

68% Deg. 

---- *" ■ ... 















































































98 



TRAVERSE 

TABLE. 



! o 
a 
a 

27 Deo. 

27% Deg. 

27 % Deg. 

.4 

27% Deo. 

vo 

5 

Lat. 

Dep. 

Lat. 

Dep. j 

Lat 

Dep. 

Lat. 

Dep. 

1 

0. 89 

0. 45 

0. 89 

0. 46 ; 

0. 89 

0. 46 

0. 83 

0. 47 

2 

1. 78 

0. 91 

1. 78 

0. 92 

1. 77 

0. 92 

1. 77 

0. 93 

3 

2. 67 

1. 36 

2. 67 

-1. 37 

2. 66 

1. 39 

2. 65 

1. 40 

4 

3. 56 

1. 82 

3. 56 

1. 83 

3. 55 

1. 85 

3. 54 

1. 86 

5 

4. 45 

2. 27 

4. 45 

2. 29 1 

4. 44 

2. 31 

4. 42 

2. 33 

1 6 

5. 35 

2. 72 

5. 33 

2. 75 

5. 32 

2. 77 

5. 31 

2. 79 

7 

6. 24 

3. 18 

6. 22 

3. 21 

6. 21 

3. 23 

6. 19 

3. 26 

8 

7. 13 

3. 63 

7. 11 

3. 66 

7. 10 

3. 69 

7. 08 

3. 72 

9 

8. 02 

4. 09 

8. 00 

4. 12 

7. 98 

4. 16 

7. 96 

4. 19 

10 

8. 91 

4. 54 

8. 89 

4. 58 

8. 87 

4. 62 

8. 85 

4. 66 

11 

9. 80 

4. 99 

9. 78 

5. 04 

9. 76 

5. 08 

9. 73 

5. 12 

12 

10. 69 

5. 45 

10. 67 

5. 49 

10. 64 

5. 54 

10. 62 

5. 59 

13 

11. 58 

5. 90 

11. 56 

5. 95 

11. 53 

6. 00 

11. 50 

6. 05 

14 

12. 47 

6. 36 

12. 45 

6. 41 

12. 42 

6. 46 

12. 39 

6. 52 

15 

13. 37 

6. 81 

13. 34 

6. 87 

13. 31 

6. 93 

13. 27 

6. 98 

16 

14. 26 

7. 26 

14. 22 

7. 33 

14. 19 

7. 39 

14. 16 

7. 45 

17 

15. 15 

7. 72 

15. 11 

7. 78 

15. 08 

7. 85 

15. 04 

7. 92 

18 

16. 04 

8. 17 

16. 00 

8. 24 

15. 97 

8. 31 

15. 93 

8. 38 

19 

16. 93 

8. 63 

16. 89 

8. 70 

16. 85 

8. 77 

16. 81 

8. 85 

20 

17. 82 

9. 08 

17. 78 

9. 16 

17. 74 

9. 23 

17. 70 

9. 31 

21 

18. 71 

9. 53 

18. 67 

9. 62 

18. 63 

9. 70 

18. 53 

9. 78 

22 

19. 60 

9. 99 

19. 56 

10. 07 

19. 51 

10. 16 

19. 47 

10. 24 

23 

20. 49 

10. 44 

20. 45 

10. 53 

20. 40 

10. 62 

20. 35 

10. 71 

24 

21. 38 

10. 90 

21. 34 

10. 99 

21. 29 

11. 08 

21. 24 

11. 17 

25 

22. 28 

11. 35 

22. 23 

11. 45 

22. 18 

11. 54 

22. 12 

11. 64 

26 

23. 17 

11. 80 

23. 11 

11. 90 

23. 06 

12. 01 

23. 01 

12. 11 

27 

24. 06 

12. 26 

24. 00 

12. 36 

23. 95 

12. 47 

23. 89 

12. 57 

28 

24. 95 

12. ,71 

24. 89 

12. 82 

24. 84 

12. 93 

24. 78 

13. 04 

29 

25. 84 

13. 17 

25. 78 

13. 28 

25. 72 

13. 39 

25. 66 

13. 50 

30 

26. 73 

13. 62 

26. 67 

13. 74’ 

26. 61 

13. 85 

26. 55 

13. 97 

35 

31. 19 

15. 89 

31. 12 

16. 03 

31. 05 

16. 16 

30. 97 

16. 30 

40 

35. 64 

18. 16 

35. 56 

18. 31 

35. 48 

18. 47 

35. 40 

18. 62 

45 

40. 10 

20. 43 

40. 01 

20. 60 

39. 92 

20. 78 

39. 82 

20. 95 

50 

44. 55 

22. 70 

44. 45 

22. 89 

44. 35 

23. 00 

1 44. 25 

23. 23 

55 

49. 01 

24. 97 

48. 90 

25. 18 

48. 79 

25. 40 

48. 67 

25. 61 

60 

53. 46 

27. 24 

53. 34 

27. 47 

53. 22 

27. 70 

53. 10 

27. 94 

65 

57. 92 

29. 51 

57. 79 

29. 76 

57. 66 

30. 01 

57. 52 

30. 26 

70 

62. 37 

31. 78 

62. 23 

32. 05 

62. 09 

32. 32 

61. 95 

32. 59 

75 

66. 83 

34. 05 

66. 68 

34. 34 

66. 53 

34. 63 

66. 37 

34. 92 

80 

71. 28 

36. 32 

71. 12 

! 36. 63 

70. 96 

36. 94 

70. 80 

37. 25 

85 

75. 74 

38. 59 

75. 57 

38. 92 

75. 40 

39. 25 

75. 22 

39. 58 

90 

80. 19 

40. 86 

80. 01 

41. 21 

79. 83 

41. 56 

79. 65 

41. 91 

95 

84. 65 

43. 13 

84. 46 

43. 50 

84. 27 

43. 87 

84. 07 

44. 23 

100 

89. 10 

45. 40 

88. 90 

45. 79 

88. 70 

46. 17 

88. 50 

46. 56 

6 

s 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lilt. 

cs 

w 

aO 

63 Deo. 

62% Deo. 

62% Deo. 

62% Deg. 




























































































TRAVERSE 

TABLE. 



99 | 

Distance. 

28 Deg. 

----- £_ 

28% Deg. 

28 % Deg. 

28 % Deo. 

B 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

| Dep. 

Lat. 

Dep. | 

1 

0. 83 

0. 

47 

0. 88 

0. 47 

0. 83 

0. 48 

0. 

88 

0. 48 

2 

1. 77 

0. 

94 

1. 76 

0. 95 

1. 76 

0. 95 

1. 

75 

0. 96 i 

3 

2. 65 

1. 

41 

2. 64 

1. 42 

2. 64 

1. 43 

2. 

63 

1. 44 

4 

3. 53 

1. 

88 

3. 52 

1. 89 

3. 52 

1. 91 

3. 

51 

1. 92 

5 

4. 41 

2. 

35 

4. 40 

2. 37 

4. 39 

2. 39 

4. 

38 

2. 40 j 

6 

5. 30 

2. 

82 

5. 29 

2. 84 

5. 27 

2. 86 

5. 

26 

2. 89 

7 

6. 18 

3. 

29 

6. 17 

3. 31 

6. 15 

3. 34 

6. 

14 

3. 37 

8 

7. 06 

3. 

76 

7. 05 

3. 79 

7. 03 

3. 82 

7. 

01 

3. 85 

9 

7. 95 

4. 

23 

7. 93 

4. 26 

7. 91 

4. 29 

7. 

89 

4. 33 

10 

8. 83 

4. 

69 

8. 81 

4. 73 

8. 79 

4. 77 

8. 

77 

4. 81 

11 

9. 71 

5. 

16 

9. 69 

5. 21 

9. 67 

5. 25 

9. 

64 

5. 29 

12 

10. 60 

5. 

63 

10. 57 

5. 68 

10. 55 

5. 73 

10. 

52 

5. 77 

13 

11. 43 

6. 

10 

11. 45 

6. 15 

11. 42 

6. 20 

| 11. 

40 

6. 25 

14 

12. 36 

6. 

57 

12. 33 

6. 63 

12. 30 

6. 68 

12. 

27 

6. 73 

15 

13. 24 

7. 

04 

13. 21 

7. 10 

13. 18 

7. 16 

! 13. 

15 

7. 21 

1G 

14. 13 

7. 

51 

14. 09 

7. 57 

14. 06 

7. 63 

1 14. 

03 

7. 70 

17 

15. 01 

7. 

98 

14. 98 

8. 05 

14. 94 

8. 11 

14. 

90 

8. 18 

13 

15. 89 

8. 

45 

15. 86 

8. 52 

15. 82 

8. 59 

15. 

78 

8. 66 

19 

16. 78 

8. 

92 

16. 74 

8. 99 

16. 70 

9. 07 

16. 

66 

9. 14 

20 

17. 66 

9. 

39 

17. 62 

9. 47 

17. 58 

9. 54 

17. 

53 

9. 62 

21 

18. 54 

9. 

86 

18. 50 

9. 94 

18. 46 

10. 02 

18. 

41 

10. 10 

22 

19. 42 

10. 

33 

19. 38 

10. 41 

19. 33 

10. 50 

19. 

29 

10. 58 

23 

20. 31 

10. 

80 

20. 26 

10. 89 

20. 21 

10. 97 

i 20. 

16 

11. 06 

24 

21. 19 

11. 

27 

21. 14 

11. 36 

21. 09 

11. 45 

; 21 . 

04 

11. 54 

25 

22. 07 

11. 

74 

22. 02 

11. 83 

21. 97 

11. 93 

21. 

92 

12. 02 

26 

22. 96 

12. 

21 

22. 90 

12. 31 

22. 85 

12. 41 

22. 

79 

12. 51 

27 

23. 84 

12. 

68 

23. 78 

12. 78 

23. 73 

12. 88 

23. 

67 

12. 99 

28 

24. 72 

13. 

15 

24. 66 

13. 25 

24. 61 

13. 36 

24. 

55 

13. 47 

29 

25. 61 

13. 

61 

25. 55 

13. 73 

25. 49 

13. 84 

25. 

43 

13. 95 

30 

20. 49 

14. 

08 

26. 43 

14. 20 

26. 36 

14. 31 

26. 

30 

14. 43 

! 35 

30. 90 

16. 

43 

30. 83 

16. 57 

30. 76 

16. 70 

30. 

69 

16. 83 

40 

35. 32 

18. 

78 

35. 24 

18. 93 

35. 15 

19. 09 

35. 

07 

19. 24 

45 

39. 73 

21. 

13 

39. 64 

21. 30 

39. 55 

21. 47 

39. 

45 

21. 64 

50 

44. 15 

23. 

47 

44. 04 

23. 67 

43. 94 

23. 86 

43. 

84 

24. 05 

55 

48. 56 

25. 

82 

48. 45 

26. 03 

48. 33 

26. 24 

48. 

22 

26. 45 

60 

52. 93 

28. 

17 

52. 85 

28. 40 

52. 73 

28. 63 

52. 

60 

28. 86 

; 65 

57. 39 

30. 

52 

57. 26 

30. 77 

57. 12 

31. 02 

56. 

99 

31. 26 

70 

61. 81 

32. 

86 

61. 66 

33. 13 

61. 52 

33. 40 

61. 

37 

33. 67 

75 

66. 22 

35. 

21 

66. 07 

35. 50 

65. 91 

35. 79 

65. 

75 

36. 07 

80 

70. 64 

37. 

56 

70. 47 

37. 87 

70. 31 

38. 17 

70. 

14 

38. 48 

85 

75. 05 

39. 

91 

74. 88 

40. 23 

74. 70 

40. 56 

74. 

52 

40. 88 

90 

79. 47 

42. 

25 

79. 28 

42. 60 

79. 09 

42. 94 

78. 

91 

43. 29 

95 

83. 88 

44. 

60 

83. 63 

44. 97 

83. 49 

45. 33 

83. 

29 

45. 69 

100 

88. 29 

46. 

95 

88. 09 

47. 33 

87. 88 

47. 72 

87. 

67 

48. 10 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

62 Deg. 

61 % Deg. 

61 % Deg. 

61 % Deg. 




















































































I 100 



TRAVERSE 

TABLE. 



(f 

1 # 

K 

1 8 

? g 

29 Deg. 

29^ 

Deg. 

29X 

Deg. 

29^ 

Deg. 

* ** 

1 

Lat. 

Dcp. 

Lat 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

1 

0. 87 

0. 48 

0. 87 

0. 49 

0. 87 

0. 49 

0. 87 

0. 50 

2 

1. 75 

0. 97 

1. 74 

0. 98 

1. 74 

0. 98 

1. 74 

0. 99 

3 

2. 62 

1. 45 

2. 62 

1. 47 

2. 61 

1. 48 

2. 60 

1. 49 

1 4 

3. 50 

1. 94 

3. 49 

1. 95 

3. 48 

1. 97 

3. 47 

1. 98 

5 

4. 37 

2. 42 

4. 36 

2. 44 

4. 35 

2. 46 

4. 34 

2. 48 

I G 

5. 25 

2. 91 

5. 23 

2. 93 

5. 22 

2. 95 

5. 21 

2. 98 | 

7 

6. 12 

3. 39 

6. 11 

3. 42 

6. 09 

3. 45 

6. 08 

3. 47 

8 

7. 00 

3. 88 

6. 98 

3. 91 

6. 96 

3. 94 

6. 95 

3. 97 

9 

7. 87 

4. 36 

7. 85 

4. 40 

7. 83 

4. 43 1 

7. 81 

4. 47 

10 

8. 75 

4. 85 

8. 72 

4. 89 

8. 70 

4. 92 , 

8. 68 

4. 96 

11 

9. 62 

5. 33 

9. 60 

5. 37 

9. 57 

5. 42 

9. 55 

5. 46 

12 

10. 50 

5. 82 

10. 47 

5. 86 

10. 44 

5. 91 

10. 42 

5. 95 

13 

11. 37 

6. 30 

11. 34 

6. 35 

11. 31 

6. 40 

11. 29 

6. 45 

14 

12. 24 

6. 79 

12. 21 

6. 84 

12. 18 

6. 89 

12. 15 

6. 95 

15 

13. 12 

7. 27 

13. 09 

7. 33 

13. 06 

7. 39 

13. 02 

7. 44 

16 

13. 99 

7. 76 

13. 96 

7. 82 

13. 93 

7. 88 

13. 89 

7. 94 

17 

14. 87 

8. 24 

14. 83 

8. 31 

14. 80 

8. 37 

14. 76 

8. 44 

18 

15. 74 

8. 73 

15. 70 

8. 80 

15. 67 

8. 86 

15. 63 

8. 93 

19 

16. 62 

9. 21 

16. 58 

9. 28 

16. 54 

9. 36 

16. 50 

9. 43 

20 

17. 49 

9. 70 

17. 45 

9. 77 

17. 41 

9. 85 

17. 36 

9. 92 

21 

18. 37 

10. 18 I 

18. 32 

10. 26 

18. 28 

10. 34 

18. 23 

10. 42 

22 

19. 24 

10. 67 

19. 19 

10. 75 

19. 15 

10. 83 

19. 10 

10. 92 

23 

20. 12 

11. 15 

20. 07 

11. 24 

20. 02 

11. 33 

19. 97 

11. 41 

24 

20. 99 

11. 64 

20. 94 

11. 73 

20. 89 

11. 82 

20. 84 

11. 91 

25 

21. 87 

12. 12 

21. 81 

12. 22 

21. 76 

12. 31 

21. 70 

12. 41 

26 

22. 74 

12. 60 

22. 63 

12. 70 

22. 63 

12. 80 

22. 57 

12. 90 

27 

23. 61 

13. 09 

23. 56 

13. 19 

23. 50 

13. 30 

23. 44 

13. 40 

28 

24. 49 

13. 57 

24. 43 

13. 68 

24. 37 

13. 79 

24. 31 

13. 89 

29 

25. 36 

14. 06 

25. 30 

14. 17 

25. 24 

14. 28 

25. 18 

14. 39 

30 

26. 24 

14. 54 

26. 17 

14. 66 

26. 11 

14. 77 

26. 05 

14. 89 

35 

30. 61 

16. 97 

30. 54 

17. 10 

30. 46 

17. 23 

30. 39 

17. 37 

40 

34. 98 

19. 39 

34. 90 

19. 54 

34. 81 

19. 70 

34. 73 

19. 85 

45 

39. 36 

21. 82 

39. 26 

21. 99 

39. 17 

i 22. 16 

39. 07 

22. 33 

50 

43. 73 

24. 24 

43. 62 

24. 43 

43. 52 

24. 62 

43. 41 

24. 81 

55 

48. 10 

26. 66 

47. 99 

26. 87 

47. 87 

27. 08 

47. 75 

27. 29 

60 

52. 48 

29. 09 

52. 35 

29. 32 

52. 22 

29. 55 

52. 09 

29. 77 

65 

56. 85 

31. 51 

56. 71 

31. 76 

56. 57 

32. 01 

56. 43 

32. 25 

70 

61. 22 

33. 94 

61. 07 

34. 20 

60. 92 

34. 47 

60. 77 

34. 74 

75 

65. 60 

36. 36 

65. 44 

36. 65 

65. 28 

36. 93 

65. 11 

37. 22 

j 80 

69. 97 

38. 78 

69. 80 

39. 09 

69. 63 

39. 39 

69. 46 

39. 70 

! 85 

74. 34 

41. 21 

74. 16 

41. 53 

73. 98 

| 41. 86 

73. 80 

42. 18 

90 

78. 72 

43. 63 

78. 52 

43. 98 

78. 33 

44. 32 

78. 14 

44. 66 

95 

83. 09 

46. 06 

82. 89 

46. 42 

82. 68 

46. 78 

82. 48 

47. 14 

100 

87. 46 

48. 48 

87. 25 

48. 86 

87. 04 

49. 24 

86. 82 

49. 62 

! | «> 
g 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

* l ■*-> 

5 

fc== 

61 

Deg. 

60M 

Deg. 

60K 

Deg. 


Deg. 































































































TRAVERSE TABLE. 101 


S3 

CJ 

30 Deo. 

30% Deg. 

30% Deg. 

30% Deg. 

C/3 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 87 

0. 50 

0. 86 

0. 50 

0. 86 

0. 51 

0. 86 

0. 51 

2 

1. 73 

1. 00 

1. 73 

1. 01 

1. 72 

1. 02 

1. 72 

1. 02 

3 

2. GO 

1. 50 

2. 59 

1. 51 

2. 58 

1. 52 

2. 58 

1. 53 

4 

3. 46 

2. 00 

3. 46 

2. 02 

3. 45 

2. 03 

3. 44 

2. 05 

5 

4. 33 

2. 50 

4. 32 

2. 52 

4. 31 

2. 54 

4. 30 

2. 56 

6 

5. 20 

3. 00 

5. 18 

3. 02 

5. 17 

3. 05 

5. 16 

3. 07 

7 

G. OG 

3. 50 

6. 05 

3. 53 

6. 03 

3. 55 

6. 02 

3. 58 

8 

G. £3 

4. 00 

6. 91 

4. 03 

6. 89 

4. 06 

6. 88 

4. 09 

9 

7. 79 

4. 50 

7. 77 

4. 53 

7. 75 

4. 57 

7. 73 

4. 60 

10 

8. GG 

5. 00 

8. 64 

5. 04 

8. 62 

5. 08 

8. 59 

5. 11 

11 

9. 53 

5. 50 

9. 50 

5. 54 

9. 48 

5. 58 

9. 45 

5. 62 

12 

10. 39 

6. 00 

10. 37 

6. 05 

10. 34 

6. 09 

10. 31 

6. 14 

13 

11. 2G 

6. 50 

11. 23 

6. 55 

11. 20 

6. 60 

11. 17 

6. 65 

14 

12. 12 

7. 00 

12. 09 

7. 05 

12. 06 

7. 11 

12. 03 

7. 16 

15 

12. 99 

7. 50 

12. 96 

7. 56 

12. 92 

7. 61 

12. 89 

7. 67 

1G 

13. 86 

8. 00 

13. 82 

8. 06 

13. 79 

8. 12 

13. 75 

8. 18 

17 

14. 72 

8. 50 

14. G9 

8. 56 

14. 65 

8. 63 

14. 61 

8. 69 

18 

15. 59 

9. 00 

15. 55 

9. 07 

15. 51 

9. 14 

15. 47 

9. 20 

19 

1G. 45 

9. 50 

16. 41 

9. 57 

16. 37 

9. 64 

16. 33 

9. 71 

20 

17. 32 

10. 00 

17. 28 

10. 08 

17. 23 

10. 15 

17. 19 

10. 23 

21 

18. 19 

10. 50 

18. 14 

10. 58 

18. 09 

10. 66 

18. 05 

10. 74 

22 

19. 05 

11. 00 

19. 00 

11. 08 

18. 96 

11. 17 

18. 91 

11. 25 

23 

19. 92 

11. 50 

19. 87 

11. 59 

19. 82 

11. 67 

19. 77 

11. 76 

24 

20. 78 

12. 00 

20. 73 

12. 09 

20. 68 

12. 18 

20. 63 

12. 27 

25 

21. 65 

12. 50 

21. 60 

12. 59 

21. 54 

12. 69 

21. 49 

12. 78 

26 

22. 52 

13. 00 

22. 4G 

13. 10 

22. 40 

13. 20 

22. 34 

13. 29 

27 

23. 33 

13. 50 

23. 32 

13. 60 

23. 26 

13. 70 

23. 20 

13. 80 

28 

24. 25 

14. 00 

24. 19 

14. 11 

24. 13 

14. 21 

24. 06 

14. 32 

29 

25. 11 

14. 50 

25. 05 

14. 61 

24. 99 

14. 72 

24. 92 

14. 83 

30 

25. 98 

15. 00 

25. 92 

15. 11 

25. 85 

15. 23 

25. 78 

15. 34 

35 

30. 31 

17. 50 

30. 23 

17. 63 

30. 16 

17. 76 

30. 08 

17. 90 

40 

34. 64 

20. 00 

34. 55 

20. 15 

34. 47 

20. 30 

84. 38 

20. 45 

45 

38. 97 

22. 50 

38. 87 

22. 67 

88. 77 

22. 84 

38. 67 

23. 01 

50 

43. 30 

25. GO 

43. 19 

25. 19 

43. 08 

25. 38 

42. 97 

25. 56 

55 

47. 63 

27. 50 

47. 51 

27. 71 

47. 39 

27. 91 

47. 27 

28. 12 

GO 

51. 96 

30. 00 

51. 83 

80. 23 

51. 70 

30. 45 

51. 56 

SO. 68 

65 

56. 29 

32. 50 

56. 15 

32. 75 

56. 01 

32. 99 

55. 86 

33. 23 

70 

60. 62 

35. 00 

60. 47 

85. 26 

60. 31 

35. 53 

60. 16 

35. 79 

75 

64. 95 

37. 50 

64. 79 

37. 78 

64. 62 

38. 07 

64. 46 

38. 85 

80 

69. 23 

40. 00 

69. 11 

40. 30 

63. 93 

40. GO 

68. 75 

40. 90 

85 

73. 61 

42. 50 

73. 43 

42. 82 

73. 24 

43. 14 

73. 05 

43. 46 

90 

77. 94 

45. 00 

77. 75 

45. 34 

77. 55 

45. 68 

77. 85 

46. 02 

95 

82. 27 

47. 50 

82. 06 

47. 88 

81. 85 

43. 22 

81. 64 

48. 57 

100 

86. 60 

50. 00 j 

86. 38 

50. 38 

86. 16 

50. 75 

85. 94 

51. 13 

• 

V 

c 

Dep. 

Lat. 

Dep. 

Lat. 

Dop. 

Lat. 

Dep. 

Lat. 

fi 

5 

60 Deg. 

59% 

Deo. 

59% Deg. 

59% Deg. 















































































102 TRAVERSE TABLE. 


4? 

O 

C 

c3 

SI Deg. 

six 

Deg. 

six 

Deg. 

31% 

- 

Deg. 

CO 

a 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

1 

0. 86 

0. 

51 

0. 85 

0. 52 

0. 85 

0. 52 

0. 85 

0. 53 

2 

1. 71 

1. 

03 

1. 71 

1. 04 

1. 71 

1. 04 

1. 70 

1. 05 

3 

2. 57 

1 . 

55 

2. 50 

1. 56 

2. 56 

1. 57 

2. 55 

1. 58 

4 

3. 43 

2. 

06 

3. 42 

2. 08 

3. 41 

2. 09 

3. 40 

2. 10* 

5 

4. 29 

2. 

58 

4. 27 

2. 59 

4. 23 

2. 61 

4. 25 

2. 63 

6 

5. 14 

3. 

09 

5. 13 

3. 11 

5. 12 

3. 13 

5. 10 

3. 16 

7 

6. 00 

3. 

61 

5. 98 

3. 63 

5. 97 

3. 66 

5. 95 

3. 68 

8 

6. 85 

4. 

12 

6. 81 

4. 15 

6. 82 

4. 18 

6. 80 

4. 21 

9 

7. 71 

4. 

64 

7. 69 

4. 67 

7. 67 

4. 701 

7. 65 

4. 74 

10 

8. 57 

5. 

15 

8. 55 

5. 19 

8. 53 

5. 22 

8. 50 

5. 26 

11 

9. 43 

5. 

67 

9. 40 

5. 71 

9. 33 

5. 75 

9. 35 

5. 79 

12 

10. 29 

6. 

18 

10. 26 

6. 23 

10. 23 

6. 27 

10. 20 

6. 31 

13 

11. 14 

6. 

70 | 

11. 11 

6. 74 

11. 08 

6. 79 

11. 05 

6. 84 

14 

12. 00 

7. 

21 1 

11. 97 

7. 26 

11. 94 

7. 31 

11. 90 

7. 37 

15 

12. 83 

7. 

73 

12. 82 

7. 78 

12. 79 

7. 84 

12. 76 

7. 80 

16 

13. 71 

8. 

24 

13. 08 

8. 30 

13. 64 

8. 36 

13. 61 

8. 42 

17 

14. 57 

8. 

76 

14. 53 

8. 82 

14. 49 

8. 88 

14. 46 

8. 95 

18 

15. 43 

9. 

27 

15. 39 

9. 34 

15. 35 

9. 40 

15. 31 

9. 47 

19 

16. 29 

9. 

79 

16. 24 

9. 86 

10 20 

9. 93 

16. 16 

10. 09 

20 

17. 14 

10. 

30 

17. 10 

10. 38 

17. 05 

10. 45 

17. 01 

10. 52 

21 

18. 00 

10. 

82 

17. 95 

10. 89 

17. 91 

10. 97 i 

17. 86 

11. 05 

22 

18. 86 

11. 

33 

18. 81 

11. 41 

18. 76 

11. 49 

18. 71 

11. 58 

23 

19. 71 

11. 

85 

19. 66 

11. 93 ! 

19. 61 

12. 02 1 

19. 56 

12. 10 

24 

20. 57 

12. 

36 

20. 52 

12. 45 

20. 46 

12. 54 

20. 41 

12. 03 

25 

21. 43 

12. 

83 

21. 37 

12. 97 

21. 32 

13. 06 

21. 20 

13. 16 

26 

22. 29 

13. 

39 

22. 23 

13. 49 i 

22. 17 

13. 58 

22. 11 

13. 63 

27 

23. 14 

13. 

91 

23. 08 

14. 01 ! 

23. 02 

14. 11 

22. 96 

14. 21 

28 

24. 00 

14. 

42 

23. 94 

14. 53 

23. 87 

14. 63 

23. 81 

14. 73 

29 

24. 86 

14. 

94 

24. 79 

15. 04 

24. 73 

15. 15 

24. 66 

15. 26 

30 

25. 71 

15. 

45 

25. 65 

15. 56 

25. 58 

15. 67 

25. 51 

15. 79 

35 

30. 00 

18. 

03 

29. 92 

18. 16 

29. 84 

18. 29 

29. 76 

18. 42 

40 

34. 29 

20. 

60 

34. 20 

20. 75 

34. 11 

20. 90 

34. 01 

21. 05 

45 

38. 57 

23. 

18 

38. 47 

23. 34 

38. 37 

23. 51 

38. 27 

23. 68 

50 

42. 86 

25. 

75 

42. 75 

25. 94 

42. 63 

26. 12 

42. 52 

26. 31 

55 

47. 14 

28. 

33 

47. 02 

28. 53 

46. 90 

I 28. 74 

46. 77 

28. 94 

60 

51. 43 

30. 

90 

51. 29 

31. 13 

51. 16 

31. 35 

51. 02 

31. 57 

65 

55. 72 

33. 

48 

55. 57 

33. 72 

55. 42 

33. 96 

55. 27 

34. 20 

70 

60. 00 

36. 

05 

59. 84 

36. 31 

59. 68 

36. 57 

59. 52 

36. 83 

75 

64. 29 

38. 

63 

64. 12 

38. 91 

63. 95 

39. 19 

63. 78 

39. 47 

80 

68. 57 

41. 

20 

68. 39 

41. 50 

68. 21 

41. 80 

68. 03 

42. 10 

85 

72. 86 

43. 

78 

72. 67 

44. 10 

| 72. 47 

44. 41 

72. 28 

44. 73 

90 

77. 15 

46. 

35 

76. 94 

46. 69 

76. 74 

47. 02 

76. 53 

47. 36 

95 

81. 43 

48. 

93 

81. 22 

49. 28 

81. 00 

49. 64 

80. 78 

49. 99 

100 

85. 72 

51. 

50 

85. 49 

51. 83 

85. 20 

52. 25 

85. 04 

52. 62 

6 

O 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

^0 

s 

59 

Deg. 


58% Deg. 

53% 

Deg. 

5S% Deg. 

) 



































































































TRAVERSE 

TABLE. 



103 

Distance. 

32 Deg. 

32% Deg. 

32% Deg. 

32% Deg. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 85 

0. 

53 

0. 85 

0. 53 

0. 84 

0. 54 

0. 

84 

0. 54 

2 

1. 70 

1. 

06 

1. 69 

1. 07 

1. 69 

1. 07 

1. 

68 

1. 08 

3 

2. 54 

1. 

59 

2. 54 

1. 60 

2. 53 

1. 61 

2. 

52 

1. 62 

4 

3. 39 

2. 

12 

3. 38 

2. 13 

3. 37 

2. 15 

3. 

36 

2. 16 

5 

4. 24 

2. 

65 

4. 23 

2. 67 

4. 22 

2. 60 

4. 

21 

2. 70 

6 

5. 09 

3. 

18 

5. 07 

3. 20 

5. 06 

3. 22 

5. 

05 

3. 25 

7 

5. 94' 

3. 

71 

5. 92 

3. 74 

5. 90 

3. 76 

5. 

89 

3. 79 

8 

6. 78 

4. 

24 

6. 77 

4. 27 

6. 75 

4. 30 

6. 

73 

4. 33 

9 

7. 63 

4. 

77 

7. 61 

4. 80 

7. 59 

4. 84 

7. 

57 

4. 87 

10 

8. 48 

5. 

30 

8. 46 

5. 34 

8. 43 

5. 37 

8. 

41 

5. 41 

11 

9. 33 

5. 

83 

9. 30 

5. 87 

9. 28 

5. 91 

9. 

25 

5. 95 

12 

10. 18 

6. 

36 

10. 15 

6. 40 

10. 12 

6. 45 

10. 

09 

6. 49 

13 

11. 02 

6. 

89 

10. 99 

6. 94 

10. 96 

6. 98 

10. 

93 

7. 03 

14 

11. 87 

7. 

42 

j 11. 84 

7. 47 

11. 81 

7. 52 

11. 

77 

7. 57 

15 

12. 72 

7. 

95 

12. 69 

8. 00 

12. 65 

8. 06 

12. 

62 

8. 11 

1G 

13. 57 

8. 

48 

13. 53 

8. 54 

13. 49 

8. 60 

13. 

46 

8. 66 

17 

14. 42 

9. 

01 

14. 38 

9. 07 

14. 34 

9. 13 

14. 

30 

9. 20 

18 

15. 26 

9. 

54 

15. 22 

9. 61 

15. 18 

9. 67 

15. 

14 

9. 74 

19 

16. 11 

10. 

07 

16. 07 

10. 14 

16. 02 

10. 21 

15. 

98 

10. 28 

20 

16. 96 

10. 

60 

16. 91 

10. 67 

16. 87 

10. 75 

i 16. 

82 

10. 82 

21 

17. 81 

11. 

13 

17. 76 

11. 21 

17. 71 

11. 28 

17. 

66 

11. 36 

22 

18. 66 

11. 

66 

18. 61 

11. 74 

18. 55 

11. 82 

18. 

50 

11. 90 

23 

19. 51 

12. 

19 

19. 45 

12. 27 

19. 40 

12. 36 

19. 

34 

12. 44 

24 

20. 35 

12. 

72 

20. 30 

12. 81 

20. 24 

12. 90 

20. 

18 

12. 98 

25 

21. 20 

13. 

25 

21. 14 

13. 34 

21. 08 

13. 43 

21. 

03 

13. 52 

26 

22. 05 

13. 

78 

21. 99 

13. 87 

21. 93 

13. 97 

21. 

87 

14. 07 

27 

22. 90 

14. 

31 

22. 83 

14. 41 

22. 77 

14. 51 

22. 

71 

14. 61 

28 

23. 75 

14. 

84 

23. 68 

14. 94 

23. 61 

15. 04 

23. 

55 

15. 15 

29 

24. 59 

15. 

37 

24. 53 

15. 47 

24. 46 

15. 58 

24. 

39 

15. 69 

30 

25. 44 

15. 

90 

25. 37 

16. 01 

25. 30 

16. 12 

25. 

23 

16. 23 

35 

29. 68 

18. 

55 

29. 60 

18. 68 

29. 52 

18. 81 

29. 

44 

18. 93 

40 

33. 92 

21. 

20 

| 33. 83 

21*. 34 

33. 74 

21. 49 

33. 

64 

21. 64 

45 

38. 16 

23. 

85 

38. 06 

24. 01 

37. 95 

24. 18 

37. 

85 

24. 34 

50 

42. 40 

26. 

50 

42. 29 

26. 68 

42. 17 

26. 86 

, 42. 

05 

27. 05 

55 

46. 64 

29. 

15 

46. 51 

29. 35 

46. 39 

29. 55 

46. 

26 

29. 75 

GO 

50. 88 

31. 

80 

50. 74 

32. 02 

50. 60 

32. 24 

50. 

46 

32. 46 

65 

55. 12 

34. 

44 

54. 97 

34. 68 

54. 82 

34. 92 

54. 

67 

35. 16 

70 

59. 36 

37. 

09 

59. 20 

37. 35 

59. 04 

37. 61 

58. 

87 

37. 87 

75 

63. 60 

39. 

74 

63. 43 

40. 02 

63. 25 

40. 30 

63. 

08 

40. 57 

80 

67. 84 

42. 

39 

67. 66 

42. 69 

67. 47 

42. 98 

67. 

28 

43. 28 

85 

72. 08 

45. 

04 

71. 89 

45. 36 

71. 69 

45. 67 

71. 

49 

45. 98 

90 

76. 32 

47. 

69 

76. 12 

48. 03 

75. 91 

48. 36 

75. 

69 

48. 69 

95 

80. 56 

50. 

34 

80. 34 

50. 69 

80. 12 

51. 04 

79. 

90 

51. 39 

100 

84. 80 

52. 

99 

84. 57 

53. 36 

84. 34 

53. 73 

84. 

10 

54. 10 

Distance. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

58 Deg. 

57% Deg. 

57% Deg. 

57% Deg. 


.1 .»• 












































































! 

104 



TRxV VERSE 

TABLE. 



6 

1 o 

g 

33 Deg. 

€>ol/ 

Deg. 

S3 % 

Deg. 

35% 

Deg. 

-4-i 

X. 

5 

Lat. 

Dcp. 

Lat. 

Dcp. 

Lat. 

Dcp. 

Lat. 

Dcp. 

1 

0. 84 

0. 54 

0. 84 

0. 55 

0. 83 

0. 55 

0. 83 

0. 56 

2 

1. 63 

1. 09 

1. 67 

1. 10 

1. 67 

1. 10 

1. 66 

1. 11 

3 

2. 52 

1. 63 

2. 51 

1. 64 

2. 50 

1. 66 

2. 49 

1. 67 

4 

3. 35 

2. 18 

3. 35 

2. 19 

3. 34 

2. 21 

3. 33 

2. 22 

5 

4. 19 

2. 72 

4. 13 

2. 74 

4. 17 

2. 76 

4. 16 

2. 78 

1 G 

5. 03 

3. 27 

5. 02 

3. 29 

5. 00 

3. 31 

4. 99 

3. 33 

7 

5. 87 

3. 81 

5. 85 

3. 84 

5. 84 

3. 86 

5. 82 

3. 89 

8 

6. 71 

4. 36 

6. 69 

4. 39 

6. 67 

4. 42 

6. 65 

4. 44 

9 

7. 55 

4. GO 

7. 53 

4. 93 

7. 50 

4. 97 

7. 48 

5. 00 

10 

8. 39 

5. 45 

8. 36 

5. 43 

8. 34 

5. 52 

8. 31 

5. 56 

11 

9. 23 

5. 99 

9. 20 

6. 03 

9. 17 

6. 07 

9. 15 

6. 11 

12 

10. 06 

6. 54 1 

10. 04 

6. 58 

10. 01 

6. 62 

9. 98 

6. 67 

13 

10. 90 

7. 08 

10. 87 

7. 13 

10. 84 

7. 13 

10. 81 

'7. 22 

14 

11. 74 

7. 62 

11. 71 

7. 63 

11. 67 

7. 73 

11. 64 

7. 78 

15 

12. 58 

8. 17 

12. 54 

8. 22 

12. 51 

8. 28 

12. 47 

8. 33 

16 

13. 42 

8. 71 

13. 38 

8. 77 

13. 34 

8. 83 

13. 30 

8. 89 

17 

14. 26 

9. 26 

14. 22 

9. 32 

14. 18 

9. 38 

14. 13 

9. 44 

1 18 

15. 10 

9. 80 

15. 05 

9. 87 

15. 01 

9. 93 

14. 97 

10. 00 

19 

15. 93 

10. 35 

15. 89 

10. 42 

15. 84 

10. 49 

15. 80 

10. 56 

20 

16. 77 

10. 89 

16. 73 

10. 97 

16. 63 

11. 04 

16. 63 

11. 11 

21 

17. 61 

11. 44 

17. 56 

11. 51 

17. 51 

11. 59 

17. 46 

11. 67 

22 

18. 45 

11. 98 

18. 40 

12. 06 

18. 35 

12. 14 

18. 29 

12. 22 

23 

19. 29 

12. 53 

19. 23 

12. 61 1 

19. 18 

12. 69 

19. 12 

12. 78 

24 

20. 13 

13. 07 

20. 07 

13. 16 

20. 01 

13. 25 

19. 96 

13. 33 

25 

20. 97 

13. 62 

20. 91 

13. 71 

20. 85 

13. 80 

20. 79 

13. 89 

26 

21. 81 

14. 16 

21. 74 

14. 26 

21. 68 

14. 35 

21. 62 

14. 44 

27 

22. 64 

14. 71 1 

22. 53 

14. 80 

22. 51 

14. 90 

22. 45 

15. 00 

28 

23. 48 

15. 25 

23. 42 

15. 35 

23. 35 

15. 45 

23. 28 

15. 56 

29 

24. 32 

15. 79 

24. 25 

15. 90 

24. 18 

16. 01 

24. 11 

16. 11 

30 

25. 16 

16. 31 

25. 09 

16. 45 

25. 02 

16. 56 

24. 94 

16. 67 

35 

29. 35 

19. 06 

29. 27 

19. 19 

29. 19 

19. 32 

29. 10 

19. 44 

40 

33. 55 

21. 79 

33. 45 

21. 93 

33. 36 

22. 08 

33. 26 

22. 22 

45 

37. 74 

24. 51 

37. 63 

24. 67 

37. 52 

24. 84 

37. 42 

25. 00 

50 

41. 93 

27. 23 

41. 81 

27. 41 

41. 69 

27. 60 

41. 57 

27. 78 

55 

46. 13 

29. 96 

46. 00 

30. 16 

45. 86 

30. 36 

45. 73 

30. 56 

60 

50. 32 

32. 68 

50. 18 

32. 90 

50. 03 

33. 12 

49. 89 

33. 33 

o5 

54. 51 

35. 40 

54. 36 

35. 64 

54. 20 

35. 88 

54. 05 

36. 11 

70 

58. 71 

38. 12 

58. 54 

33. 38 

58. 37 

38. 64 

58. 20 

38. 89 

75 

62. 90 

40. 85 

62. 72 

41. 12 

62. 54 

41. 40 

62. 36 

41. 67 

80 

67. 09 

43. 57 

66. 90 

43. 86 

66. 71 

44. 15 

66. 52 

44. 45 

85 

71. 29 

46. 29 

71. 08 

46. 60 

70. 88 

46. 91 

70. 67 

47. 22 

90 

75. 48 

49. 02 

75. 27 

49. 35 

75. 05 

49. 67 

74. 83 

50. 00 

i 95 

79. 67 

51. 74 

79. 45 

52. 09 

79. 22 

52. 43 

78. 99 

52. 78 

<100 

_ 

83. 87 

54. 46 

83. 63 

54. 83 

83. 39 

; 55. 19 

83. 15 

| 55. 56 

o 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

c3 

-4-> 

I ^ 

5 

57 

Deg. 

56 % 

Deg. 

56% Deg. 

36 % 

Deg. 







































































































TRAVERSE 

TABLE. 


105 1 

. BJ 

6 

c? 

G 

c3 

34 Deg. 

54)4 Deg. 

34% Deg. 

34% Deg. > 

5 

Lat. 

Dcp. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep 

1 

0. 83 

0. 58 

0. 83 

0. 56 

0. 82 

0. 57 

0. 82 

0. 57 

2 

1. 68 

1. 12 

1. 65 

1. 13 

1. 65 

1. 13 

1. 64 

1. 14 

3 

2. 49 

1. 68 

2, 48 

1. 69 

2. 47 

1. 70 

2, 46 

1. 71 

4 

3. 32 

2. 24 

3. 31 

2. 25 

3. 30 

2. 27 

3. 29 

2. 28 

5 

4. 15 

2. 80 

4. 13 

2. 81 

4. 12 

2. 83 

4. 11 

2. 85 

G 

4. 97 

3. 36 

4. 86 

3. 38 

4. 94 

3. 40 

4. 93 

3. 42 

7 

5. 80 

3. 91 

5. 79 

3. 94 

5. 77 

3. 86 

5. 75 

3. 99 

8 

6. 63 

4. 47 

6. 61 

4. 50 

6. 59 

4. 53 

6. 57 

4. 56 

9 

7. 46 

5. 03 

7. 44 

5. 07 

7. 42 

5. 10 

7. 39 

5. 13 

10 

8. 29 

5. 59 

8. 27 

5. 63 

8. 24 

5. 66 

8. 22 

5. 70 

11 

9. 12 

6. 15 

9. 09 

6. 19 

9. 07 

6. 23 

9. 04 

6. 27 

12 

9. 95 

6. 71 

9. 92 

6. 75 

9. 89 

6. 80 

9. 86 

6. 84 

13 

10. 78 

7. 27 

10. 75 

7. 32 

10. 71 

7. 36 

10. 68 

7. 41 

14 

11. 61 

7. 83 

11. 57 

7. 88 

11. 54 

7. 93 

11. 50 

7. 98 

15 

12. 44 

8. 39 

12. 40 

8. 44 

12. 36 

8. 50 

12. 32 

8. 55 

10 

13. 26 

8. 95 

13. 23 

9. 00 

13. 19 

9. 06 

13. 15 

9. 12 

17 

14. 09 

9. 51 

14. 05 

9. 57 

14. 01 

9. 63 

13. 87 

9. 69 ; 

18 

14. 92 

10. 07 

14. 88 

10. 13 

14. 83 

10. 20 

14. 79 

10. 26 ; 

19 

15. 75 

10. 62 

15. 71 

10. 69 

15. 66 

10. 76 

15. 61 

10. 83 

20 

16. 58 

11. 18 

16. 53 

11. 26 

16. 48 

11. 33 

16. 43 

11. 40 

21 

17. 41 

11. 74 

17. 36 

11. 82 

17. 31 

11. 89 

17. 25 

11. 97 ! 

22 

18. 24 

12. 30 

18. 18 

12. 38 

18. 13 

12. 46 

18. 08 

12. 54 

23 

19. 07 

12. 86 

19. 01 

12. 94 

18. 95 

13. 03 

18. 90 

13. 11 

24 

19. 80 

13. 42 

19. 84 

13. 51 

19. 78 

13. 59 

19. 72 

13. 68 

25 

20. 73 

13. 88 

20. 66 

14. 07 

20. 60 

14. 16 

20. 54 

14. 25 

26 

21. 55 

14. 54 

21. 49 

14. 63 

21. 43 

14. 73 

21. 86 

14. 82 

27 

22. 38 

15. 10 

22. 32 

15. 20 

22. 25 

15. 29 

22. 18 

15. 39 

28 

23. 21 

15. 66 

23. 14 

15. 76 

23. 08 

15. 86 

23. 01 

15. 96 

29 

24. 04 

16. 22 

23. 97 

16. 32 

23. 80 

16. 43 

23. 83 

16. 53 

30 

24. 87 

16. 78 

24. 80 

16. 88 

24. 72 

16. 99 

24. 65 

17. 10 

35 

29. 02 

19. 57 

28. 93 

19. 70 

28. 84 

19. 82 

28. 76 

19. 95 

40 

33. 16 

22. 37 

33. 06 

22. 51 

32. 97 

22. 66 

32. 87 

22. 80 

45 

37. 31 

25. 16 

37. 20 

25. 33 

37. 09 

25. 49 

36. 97 

25. 65 

50 

41. 45 

27. 26 

41. 33 

28. 14 

41. 21 

28. 32 

41. 08 

28. 50 

55 

45. 60 

30. 76 

45. 46 

30. 95 

45. 83 

31. 15 

45. 19 

31. 35 I 

GO 

49. 74 

33. 55 

49. 60 

33. 77 

49. 45 

33. 98 

49. 30 

34. 20 

65 

53. 89 

36. 35 

53. 73 

36. 58 

53. 57 

36. 82 

53. 41 

37. C5 

| 70 

58. 03 

39. 14 

57. 86 

39. 40 

57. 69 

39. 65 

57. 52 

39. 20 

75 

62. 18 

41. 94 

61. 99 

42. 21 

61. 81 

42. 48 

61. 62 

42. 75 

80 

66. 32 

44. 74 

66. 13 

45. 02 

65. 93 

45. 31 

65. 73 

45. 60 

85 

70. 47 

47. 53 

70. 26 

47. 84 

70. 05 

48. 14 

69. 84 

48. 45 

90 

74. 61 

50. 33 

74. 39 

50. 65 

74. 17 

50. 98 

73. 95 

51. 30 

95 

78. 76 

53. 12 

78. 53 

53. 47 

78. 29 

53. 81 

78. 06 

54. 15 

100 

82. 90 

55. 92 

82. 66 

56. 28 

82. 41 

56. 64 

82. 16 

57. 00 

oJ 

Dep. 

Lat. 

Dep. 

L:it. 

Dep. 

Lat. 

Dep. 

LaL 

1 

n 

5 

56 Deg. 

5 o% Deg. 

55% Deg. 

55% Deg. | 

- - . — 































































































106 



TRAVERSE 

TABLE. 



O 

o 

g 

cS 

35 Deg. 

S5X 

Deg. 

35% 

Deg. 

35% 

Deg. 

to 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 82 

0. 57 

0. 82 

0. 58 

0. 81 

0. 58 | 

0. 81 

0. 58 

2 

1. 64 

1. 15 

1. 63 

1. 15 

1. 63 

1. 16 

1. 62 

1. 17 

3 

2. 46 

1. 72 

2. 45 

1. 73 

2. 44 

1. 74 

2. 43 

1. 75 

4 

3. 28 

2. 29 

3. 27 

2. 31 

3. 26 

2. 32 

3. 25 

2. 34 

5 

4. 10 

2. 87 

4. 08 

2. 89 

4. 07 

2. 90 

4. 06 

2. 92 \ 

6 

4. 91 

3. 41 

4. 90 

3. 46 

4. 88 

3. 48 

4. 87 

3. 51 

7 

5. 73 

4. 01 

5. 72 

4. 04 

5. 70 

4. 06 

5. 68 

4. 09 

8 

6. 55 

4. 59 

6. 53 

4. 62 

6. 51 

4. 65 

6. 49 

4. 67 

9 

7. 37 

5. 16 

7. 35 

5. 19 

7. 33 

5. 23 

7. 30 

5. 26 

10 

8. 19 

5. 74 

8. 17 

5. 77 

8. 14 

5. 81 

8. 12 

5. 84 

11 

9. 01 

6. 31 

8. 98 

6. 35 

8. 96 

6. 39 

8. 93 

6. 43 

12 

9. 83 

6. 83 

9. 80 

6. 93 

9. 77 

6. 97 

9. 74 

7. 01 

13 

10. 65 

7. 46 

10. 62 

7. 50 

10. 58 

7. 55 

10. 55 

7. 60 

14 

11. 47 

8. 03 

11. 43 

8. 08 

11. 40 

8. 13 

11. 36 

8. 18 

15 

12. 29 

8. 60 | 

12. 25 

8. 66 

12. 21 

8. 71 

12. 17 

8. 76 

16 

13. 11 

9. 18 I 

13. 07 

9. 23 

13. 03 

9. 29 

12. 99 

9. 35 

17 

13. 93 

9. 75 

13. 88 

9. 81 

13. 84 

9. 87 

13. 80 

9. 93 

18 

14. 74 

10. 32 

14. 70 

10. 39 

14. 65 

10. 45 

14. 61 

10. 52 

19 

15. 56 

10. 90 

15. 52 

10. 97 

15. 47 

11. 03 

15. 42 

11. 10 

20 

16. 38 

11. 47 

16. 33 

11. 54 

16. 28 

11. 61 

16. 23 

11. 68 

21 

17. 20 

12. 05 

17. 15 

12. 12 

17. 10 

12. 19 

17. 04 

12. 27 

22 

18. 02 

12. 62 

17. 97 

12. 70 

17. 91 

12. 78 

17. 85 

12. 85 

23 

18. 84 

13. 19 

18. 78 

13. 27 

18. 72 

13. 36 

18. 67 

13. 44 

24 

19. 68 

13. 77 

19.. 60 

13. 85 

19. 54 

13. 94 

19. 48 

14. 02 

25 

20. 48 

14. 34 

20. 42 

14. 43 

20. 35 

14. 52 

20. 29 

14. 61 

26 

21. 30 

14. 91 | 

21. 23 

15. 01 

21. 17 

15. 10 

21. 10 

15. 19 

27 

22. 12 

15. 49 ! 

22. 05 

15. 58 

21. 98 

15. 68 

21. 91 

15. 77 

23 

22. 94 

16. 06 ! 

22. 87 

16. 16 

I 22. 80 

16. 26 

22. 72 

1C. 36 

29 

23. 76 

16. 63 i 

23. 68 

16. 74 

j 23. 61 

16. 84 

23. 54 

16. 94 

30 

24. 57 

17. 21 

24. 50 

17. 31 

24. 42 

17. 42 

24. 35 

17. 53 

35 

28. 67 

20. 03 ! 

28. 58 

20. 20 

28. 49 

20. 32 

28. 41 

20. 45 

40 

32. 77 

22. 94 

32. 67 

23. 09 

32. 56 

23. 23 

32. 46 

23. 3? 

45 

36. 86 

25. 81 

36. 75 

25. 97 

36. 64 

26. 13 

36 52 

26. 29 

50 

40. 96 

28. 68 

40. 83 

28. 86 

40. 71 

29. 04 

40. 58 

29. 21 

55 

45. 05 

31. 55 

44. 92 

31. 74 

44. 78 

31. 94 

44. 64 

32. 13 

60 

49. 15 

31. 41 

49. 00 

34. 63 

48. 85 

34. 84 

48. 69 

35. 05 

65 

53. 24 

37. 28 

53. 03 

37. 51 

52. 92 

37. 75 

52. 75 

37. 93 

70 

57. 34 

40. 15 

57. 16 

40. 40 

56. 99 

40. 65 

56. 81 

40. 90 

75 

61. 44 

43. 02 

61. 25 

43. 29 

61. 06 

43. 55 

60. 87 

43. 82 

80 

65. 53 

45. 89 

65. 33 

46. 17 

65. 13 

46. 46 

64. 93 

46. 74 j 

85 

69. 63 

48. 75 

69. 41 

49. 06 

69. 20 

49. 36 

68. 98 

49. 66 

90 

73. 72 

51. 62 

73. 50 

51. 94 

73. 27 

52. 26 

73. 04 

52. 53 

95 

77. 82 

54. 49 

77. 58 

54. 83 

77. 34 

55. 17 

77. 10 

55. 50 

100 

81. 92 

57. 36 

81. 66 

57. 71 

81. 41 

58. 07 

81. 16 

58. 42 

V 

o 

G 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

n 

"jo 

1 5 

55 Deg. 

54% 

Deg. 

54% 

Deg. 

54% Deg. 

































































































traverse 

TABLE. 


107 

4) 

O 

0 

a 

S6 Deg. 

3€% Deo. 

36% Deg. 

86% Deo. 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Di p. 

1 

0. 81 

0. 59 

0. 81 

0. 59 

0. 80 

0. 59 

0. 80 

0. 60 

2 

1. 62 

1. 18 

1. 61 

1. 18 

1. 61 

1. 19 

1. 60 

1. 20 

3 

2. 43 

1. 76 

2. 42 

1. 77 

2 41 

1. 78 

2. 40 

1. 79 

4 

3. 24 

2. 35 

3. 23 

2. 37 

3. 22 

2. 38 

3. 20 

2. 39 

5 

4. 05 

2 94 

4. 03 

2. 96 

4. 02 

2. 97 

4. 01 

2. 99 

6 

4. 85 

3. 53 

4. 84 

3. 55 

4. 82 

3. 57 

4. 81 

3. 59 

7 

6. 66 

4. 11 

5. 65 

4. 14 

5. 63 

4. 16 

5. 61 

4. 19 

8 

6. 47 

4. 70 

6. 45 

4. 73 

6. 43 

4. 76 

6. 41 

4. 79 

0 

7. 28 

5. 29 

7. 26 

5. 32 

. 23 

5. 35 

7. 21 

5. 38 

10 

8. 09 

5. 88 

8. 06 

5 91 

8. 04 

5. 95 

8. 01 

5. 98 

1 11 

8. 90 

6. 47 

8. 87 

6. 50 

8. 84 

6. 54 

8. 81 

6 58 

! 12 

9. 71 

7. 05 

9. 68 

7. 10 

9. 65 

7. 14 

9. 61 

7. 18 

13 

10. 52 

7. 64 

10. 48 

7. 69 

10. 45 

7. 73 

10. 42 

7. 78 

1 14 

11. 33 

8. 23 

11. 29 

8. 28 

11. 25 

8. 33 

11. 22 

8. 38 

15 

12. 14 

8. 82 

12. 10 

8. 87 

12. 06 

8. 92 

12. 02 

8. 97 

16 

12. 94 

9. 40 

12. 90 

9. 46 

12. 86 

9. 52 

12. 82 

9. 57 

17 

13. 7 ) 

9. 99 

13. 71 

10. 05 

13. 67 

10. 11 

13. 62 

10. 17 

18 

14. 56 

10. 58 

14. 52 

10. 64 

14. 47 

10. 71 

14. 42 

10. 77 

19 

15. 37 

11. 17 

15. 32 

11. 23 

15. 27 

11. 30 

15. 22 

11. 37 

20 

16. 18 

11. 76 

16. 13 

11. 83 

16. 08 

11. 90 

16. 03 

11. 97 

21 

16. 99 

12. 34 

16. 94 

12. 42 

16. 88 

12. 49 

.16. 83 

12. 56 

22 

17. 80 

12. 93 

17. 74 

13. 01 

17. 68 

13. 09 

17. 63 

13. 16 

23 

18. 61 

13. 52 

18. 55 

13. 60 

18. 49 

13. 68 

18. 43 

13. 76 

24 

19. 42 

14. 11 

19. 35 

14. 19 

19. 29 

14. 28 

19. 23 

14. 36 

25 

20. 23 

14. 69 

20. 16 

14. 78 

20. 10 

14 87 

20. 03 

14. 96 

26 

21. 03 

15. 28 

20. 97 

15. 37 

20. 90 

15. 47 

20 83 

15. 56 

27 

21. 84 

15. 87 

21. 77 

15. 97 

21. 70 

13. 06 

21. 63 

16. 15 

28 

22. 65 

16. 46 

22. 58 

16. 56 

22. 51 

16. 65 

22. 44 

16. 75 

29 

23. 46 

17. 05 

23. 39 

17. 15 

23. 31 

17. 25 

23. 24 

17. 35 

30 

24. 27 

17. 63 

24. 19 

17. 74 

24. 12 

17. 84 

24. 04 

17. 95 

35 

28. 32 

20. 57 

28. 23 

20. 70 

28. 13 

20. 82 

28. 04 

20. 94 

40 

32. 36 

23. 51 

32. 26 

23. 65 

32. 15 

23. 79 

32 05 

23. 93 

45 

36. 41 

26. 45 

36. 29 

26. 61 

36. 17 

26. 77 

36. 06 

26. 92 

50 

40. 45 

29. 39 

40. 32 

29. 57 

40. 19 

29. 74 

40. 06 

29. 92 

55 

44. 50 

32. 33 

44. 35 

32. 52 

44. 21 

32. 72 

44. 07 

32. 91 

60 

48. 54 

35. 27 

48. 39 

35. 48 

48. 23 

35. 69 

48. 08 

35. 9(» 

65 

52. 59 

38. 21 

52. 42 

38. 44 

52. 25 

38. 66 

52. 08 

38. 89 

70 

56. 63 

41. 14 

56. 45 

41. 39 

56. 27 

41. 64 

56. 09 

41. 88 

75 

60. 68 

44. 08 

60. 48 

44. 35 

60. 29 

44. 61 

60. 09 

44. 87 

80 

64. 72 

47. 02 

64. 52 

47. 30 

64. 31 

47. 59 

64. 10 

47. 87 

85 

68. 77 

49. 96 

68. 55 

50. 26 

68. 33 

50. 56 

68. 11 

50 86 

90 

72. 81 

52. 90 

72. 58 

53. 22 

72. 35 

53. 53 

72. 11 

53. 85 

95 

76. 86 

55. 84 

76. 61 

56. 17 

76. 37 

56. 51 

76. 12 

56. 84 

100 

80. 90 

58. 78 

80. 64 

59. 13 

80. 39 

59. 48 

80. 13 

59. 83 

i 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

§ 

CO 

5 

84 Deg. 

53% Deo. 

53)4 Deo. 

53% Deg. 






































































108 



TRAVERSE 

TABLE. 


-1 

© 

o 

a 

cS 

37 Dkg. 

37% Deg. 

37% Dbg. 

37% Ibg. 

00 

a 

Lat. 

Dep. 

J.at. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 80 

0. 60 

0. 80 

0. 61 

0. 79 

0. 61 

0. 79 

0. 61 

2 

1. 60 

1. 20 

1. 59 

1. 21 

1. 59 

1. 22 

1. 58 

1. 22 

3 

2. 40 

1. 81 

2 39 

1. 82 

2. 38 

1. 83 

2. 37 

1. 84 

4 

3. 19 

2. 41 

3. 18 

2. 42 

3. 17 

2. 43 

3. 16 

2. 45 

5 

3. 99 

8. 01 

3. 98 

3. 03 

3. 97 

3. 04 

3. 95 

3. 06 

0 

4. 79 

3. 61 

4. 78 

3. 63 

4. 76 

3. 65 

4. 74 

3. 67 

7 

5. 59 

4. 21 

5. 57 

4. 24 

5. 55 

4. 26 

5. 53 

4. 29 

8 

6. 39 

4. 81 

6. 37 

4. 84 

6. 35 

4. 87 

6. 33 

4. 90 

9 

7. 19 

5. 42 

7. 16 

5. 45 

7. 14 

5. 48 

7. 12 

5. 51 

10 

7. 99 

•6. 02 

7. 96 

6. 05 

7. 93 

6. 09 

7. 91 

6. 12 

11 

8. 78 

6. 62 

8. 76 

6. 66 

8. 73 

6. 70 

8. 70 

6. 73 

12 

9. 58 

7. 22 

9. 55 

7. 26 

9. 52 

7. 31 

9. 49 

7. 35 

13 

10. 38 

7. 82 

10. 35 

7. 87 

10. 31 

7. 91 

10. 28 

7. 96 

14 

11. 18 

8. 43 

11. 14 

8. 47 

11. 11 

8. 52 

11. 07 

8. 57 

15 

11. 98 

9. 03 

11. 94 

9. 0^ 

11. 90 

9. 13 

11. 86 

9. 18 

16 

12. 78 

9. 63 

12. 74 

9. 68 

12. 69 

9. 74 

12. 65 

9. 80 

17 

13. 58 

10. 28 

13. 53 

10. 29 

13. 49 

10. 35 

13. 44 

10. 41 

18 

14. 3S 

10. 83 

14. 33 

10. 90 

14. 28 

10. 96 

14. 23 

11. 02 

19 

15. 17 

11. 43 

15. 12 

11. 50 

15. 07 

11. 57 

15. 02 

11. 63 

20 

15. 97 

12. 04 

15. 92 

12. 11 

15. 87 

12. 18 

15. 81 

12. 24 

21 

16. 77 

12. 64 

16. 72 

12. 71 

16. 66 

12. 78 

16 60 

12. 86 

22 

17. 57 

13. 24 

17. 51 

13. 32 

17. 45 

13. 39 

17. 40 

13. 47 

23 

18. 37 

13. 84 

18. 31 

13. 92 

18. 25 

14. CO 

18. 19 

14. OS 

24 

19. 17 

14. 44 

19. 10 

14 53 

19. 04 

14. 61 

18. 98 

14. 69 

25 

19. 97 

15. 05 

19. 90 

15. 13 

19. 83 

15. 22 

19. 77 

15. 31 

26 

20. 76 

15. 65 

20. 70 

15. 74 

20. 63 

15. 83 

20. 56 

15. 92 

27 

21. 56 

16. 25 

21. 49 

16. 34 

21. 42 

16. 44 

21. 35 

16. 53 

28 

22. 36 

16. 85 

22. 29 

16. 95 

22 21 

17. 05 

22. 14 

17. 14 

29 

1 

23. 16 

17. 45 

23. 08 

17. 55 

23. 01 

17. 65 

22, 93 

17. 75 

l 

30 

23. 96 

18. 05 

23. 88 

18. 16 

23. 80 

18. 26 

23. 72 

18. 37 

! 85 

27. 95 

21. 06 

27. 86 

21. 19 

27. 77 

21. 31 

27. 67 

21. 43 

40 

31. 95 

24. 07 

31. 84 

24. 21 

31. 73 

24. 35 

31. 63 

24. 49 

45 

35. 94 

27. 08 

35. 82 

27. 24 

35. 70 

27. 39 

35. 58 

27. 55 

1 50 

39. 93 

30. 09 

39. 80 

30. 26 

39. 67 

30. 44 

39. 53 

30. 61 

! 55 

43. 92 

33. 10 

43. 78 

33. 29 

43. 63 

33. 48 

43. 49 

33. 67 

60 

47. 92 

36. 11 

47. 76 

36. 32 

47. 60 

36. 53 

47. 44 

36. 73 

, 65 

51. 91 

39. 12 

51. 74 

39. 34 

51. 57 

39. 57 

51. 39 

39. 79 

1 70 

55. 90 

42. 13 

55. 72 

42. 37 

55. 53 

42. 61 

55. 35 

42. 86 

75 

59. 90 

45. 14 

59. 70 

45. 40 

59. 50 

45. 66 

59. 30 

45 92 

80 

63. 89 

48. 15 

63. 68 

48. 42 

63. 47 

48. 70 

63. 26 

48. 98 

85 

67. 88 

51. 15 

67. 66 

51. 45 

67. 43 

51. 74 

67. 21 

52. 04 

90 

71. 88 

54. 16 

71. 64 

54. 48 

71. 40 

54. 79 

71. 16 

55. 10 

95 

75. 87 

57. 17 

75. 62 

57. 50 

75. 37 

57. 83 

75. 12 

58. 16 

100 

79. 86 

60. 18 

79. 60 

60. 53 

79. 34 

60. 88 

79. 07 

61. 22 

• 

o 

3 

m 

5 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

53 Deo. 

52 X Deo. 

52% Deo. 

52% Deo. 




















































TRAVERSE 

1 

TABLE. 



109 l 

Distance. 

38 1)eq. 

88% Deg. 

88% Deg. 

i 

38% 

Deg. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 79 

0. 62 

0. 79 

0. 62 

0. 78 

0. 

62 

0. 78 

0. 63 

2 

1. 58 

1. 23 

1. 57 

1. 24 

1. 57 

1. 

24 

1. 56 

1. 25 

3 

2. 36 

1. 85 

2. 36 

1. 86 

2. 35 

1. 

87 

2. 34 

1. 88 

4 

3. 15 

2. 46 

3. 14 

2. 48 

3. 13 

2. 

49 

3. 12 

2. 50 

5 

3. 94 

3. 08 

3. 93 

3. 10 

3. 91 

3. 

11 

3. 90 

3. 13 

6 

4. 73 

3. 69 

4. 71 

3. 71 

4. 70 

3. 

74 

4. 68 

3. 76 

7 

5. 52 

4. 31 

5. 50 

4. 33 

5. 48 

4. 

36 

5. 46 

4. 38 

8 

6. 30 

4. 93 

6. 28 

4. 95 

6. ‘26 

4. 

98 

6 21 

5. 01 

9 

7. 09 

5. 54 

7. 07 

5. 57 

7. 0 4 

5. 

60 

7. 02 

5. 63 

10 

7. 88 

6. 16 

7. 85 

6. 19 

7. 83 

6. 

23 

7. 80 

6. 26 

11 

8. 67 

6. 77 

8. 64 

6. 81 

8. 61 

6. 

85 

8. 58 

6. 89 

12 

9. 46 

7. 39 

9. 42 

7 43 

9. 39 

7. 

47 

9. 36 

7. 51 

13 

10. 24 

8. 00 

10. 21 

8. 05 

10. 17 

8 

09 

10. 14 

8. 14 

14 

11. 03 

8. 62 

in. 99 

8. 67 

10. 96 

8. 

72 

10. 92 

8. 76 

15 

11. 82 

9. 23 

11. 78 

9. 29 

11. 74 

9. 

31 

11. 70 

9. 39 

16 

12. 61 

9. 85 

12 57 

9. 91 

12. 52 

9. 

96 

12. 48 

10. 01 

17 

13. 40 

10. 47 

13. 35 

10. 52 

13. 30 

10. 

58 

13. 26 

10. 64 

18 

14. 18 

11. 08 

14. 14 

11. 14 

14. 09 

11. 

21 

14. 04 

11. 27 

19 

14. 97 

11. 70 

14. 92 

11. 76 

14. 87 

11. 

83 

14 82 

11. 89 

| 20 

15. 76 

12. 31 

15. 71 

12. 38 

15. 65 

12. 

45 

15. 60 

12. 52 

21 

16. 55 

12 93 

16. 49 

13. 00 

16. 43 

13. 

<•7 

16. 38 

13. 14 

22 

17. 34 

13. 54 

17. 28 

13. 62 

17. 22 

13. 

70 

17. 16 

13. 77 

23 

18. 12 

14. 16 

18. 06 

14. 24 

18. 00 

14. 

32 

17. 94 

14. 40 

24 

18. 91 

14. 78 

18. 85 

14. 86 

18. 78 

14. 

94 

18. 72 

15. 02- 

25 

19. 70 

15. 39 

19. 63 

15. 48 

19. 57 

15. 

56 

19. 5 ) 

15. 65 

26 

20. 49 

16. 01 

20. 42 

16. 10 

20. 35 

16. 

19 

20. 28 

16. 27 

27 

21. 28 

16. 62 

21. 20 

16. 72 

21. 13 

16. 

81 

21. 06 

16. 90 

28 

22. 06 

17. 24 

21. 99 

17. 33 

21. 91 

17. 

43 

21. 84 

17. 53 

29 

22. 85 

17. S5 

22. 77 

17. 95 

22. 70 

18. 

05 

22. 62 

18. 15 

30 

23. 64 

18. 47 

23. 56 

18. 57 

23. 48 

18. 

68 

23. 40 

18 78 

35 

27. 58 

21. 55 

27. 49 

21. 67 

27. 39 

21. 

79 

27. 30 

21. 91 

40 

31. 52 

24. 63 

31. 41 

24. 79 

31. 30 

24. 

90 

31. 20 

25. 04 

45 

35. 46 

27. 70 

35. 34 

27. 86 

35. 22 

28. 

01 

35. 09 

28. 17 

50 

39. 40 

30. 78 

39. 27 

30. 95 

39. 13 

31. 

13 

38. 99 

31. 30 

55 

43. 34 

33. 86 

43. 19 

34. 05 

43. 04 

34. 

24 

42. 89 

34. 43 

60 

47. 28 

36. 94 

47. 12 

'37. 15 

46. 96 

37. 

35 

46. 79 

37. 56 

65 

51. 22 

40. 02 

51. 05 

40. 24 

50. 87 

40. 

46 

50. 69 

40. 68 

70 

55. 16 

43. 10 

54. 97 

43. 34 

54. 78 

43. 

58 

54. 59 

43. 81 

75 

59. 10 

46. 17 

58. 90 

46. 43 

58. 70 

46. 

69 

58. 49 

46. 94 

80 

63. 04 

49. 25 

62. 83 

49. 53 

62. 61 

49. 

80 

62. 39 

50. 07 

85 

66. 98 

52. 33 

66. 75 

52. 62 

66. 52 

52. 

91 

66. 29 

53. 20 

90 

70. 92 

55. 41 

70. 68 

55. 72 

70. 43 

56. 

03 

70. 19 

56. 33 

95 

74. 86 

58. 49 

74. 61 

58. 81 

74. 35 

59. 

14 

74. 09 

59. 46 

100 

78. 80 

61. 57 

78. 53 

61. 91 

78. 26 

62. 

25 

77. 99 

62. 59 

4 

S3 

* 

« 

5 1 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

52 11 go. 

61 X Deo. 

tl% Deo. 


61% Dk«. 



































































I 110 TRAVERSE TABLE. 


9 

O 

B 

39 Deo. 

39# Deo. 

39# 

Deo. 

39# 

Deg. 


CO 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 78 

0. 61 

0. 77 

0. 63 

0. 77 

0. 64 

0. 77 

0. 

64 

2 

1. 55 

1. 26 

1. 55 

1. 27 

1. 54 

1. 27 

1. 54 

1. 

28 

3 

2. 33 

1. 89 

2. 32 

1. 90 

2. 31 

1. 91 

2. 31 

1. 

92 

4 

3. 11 

2. 52 

3. 10 

2. 53 

3. 09 

2. 54 

3. 08 

2. 

56 i 

5 

3. 89 

3. 15 

3. 87 

3. 16 

3. 86 

3. 18 

3. 84 

3. 

20 

6 

4. 66 

3. 78 

4. 65 

3. 80 

4. 63 

3. 82 

4. 61 

3. 

84 

7 

5. 44 

4. 41 

5. 42 

4. 43 

5. 40 

4. 45 

5. 38 

4. 

48 

8 

6. 22 

5. 03 

6. 20 

5. 06 

6. 17 

5. 09 

6. 15 

5. 

12 , 

9 

6. 99 

5. 60 

6. 97 

5. 69 

6. 9 4 

5. 72 

6. 92 

5. 

75 

10 

7. 77 

6. 29 

7. 74 

6 33 

7. 72 

6. 36 

7. 69 

6. 

39 

11 

8. 55 

6. 92 

8. 52 

6. 96 

8. 49 

7. 00 

8. 46 

7. 

03 

12 

9. 33 

7. 55 

9. 29 

7. 59 

9. 26 

7. 63 

9. 23 

7. 

67 

13 

10. 10 

8. 18 

10. 07 

8. 23 

10. 03 

8. 27 

9. 99 

8. 

31 

14 

10. 88 

8. 81 

10. 84 

8. 86 

10. 80 

8. 91 

10. 76 

8. 

95 

15 

11. 66 

9. 44 

11. 62 

9. 49 

11. 57 

9. 54 

11. 53 

9. 

59 

10 

12. 43 

10. 07 

12. 39 

10. 12 

12. 35 

10. 18 

12. 30 

10. 

23 

17 

13. 21 

10. 70 

13. 16 

10. 76 

13. 12 

10. 81 

13. 07 

10. 

87 

18 

13. 99 

11. 33 

13. 94 

11. 39 

13. 89 

11. 45 

13. 84 

11. 

51 ! 

19 

14. 77 

11. 96 

14. 71 

12. 0. 

14. 66 

12. 0.9 

14. 61 

12. 

15 

20 

15. 54 

12. 59 

15. 49 

12. 65 

15. 43 

12. 72 

15. 38 

12. 

79 

21 

16. 32 

13. 22 

16. 26 

13. 29 

16. 20 

13. 36 

16. 15 

13. 

43 

22 

17. 10 

13. 84 

17. 04 

13. 92 

16. 98 

13. 99 

16. 91 

14. 

07 

23 

17. 87 

14. 47 

17. 81 

14. 55 

17. 75 

14. 63 

17. 68 

14. 

71 

24 

18. 65 

15. 10 

18. 59 

15. 18 

18. 52 

15. 27 

18. 45 

15. 

35 

25 

19. 43 

15. 73 

19. 36 

15. 82 

19. 29 

15. 9 J 

19. 22 

15. 

99 

26 

20. 21 

16. 36 

20. 13 

16. 45 

20. 03 

16. 54 

19. 99 

16. 

63 

27 

20. 93 

16. 99 

20. 91 

17. 08 

2 ». 83 

17. 17 

20. 76 

17. 

26 

28 

21. 76 

17. 62 

21. 68 

17 72 

21. 61 

17. 81 

21. 53 

17. 

9o 

29 

22. 54 

18. 25 

22. 46 

18. 35 

22. 38 

18. 45 

22. 30 

18. 

54 | 

30 

23. 31 

18. 88 

23. 23 

18. 98 

23. 15 

19. 08 

23. 07 

19 

18 

35 

27. 20 

22. 03 

27. 10 

22. 14 

27. 01 

22. 26 

26. 91 

22. 

38 

40 

31. 09 

25. 17 

30. 98 

25. 31 

30. 86 

25. 44 

30. 75 

25. 

58 

45 

34. 97 

28. 32 

34. 85 

28. 47 

34. 72 

28. 62 

34. 60 

28. 

77 * 

50 

38. 86 

31. 47 

38. 72 

31. 64 

38. 58 

31. 80 

38. 44 

31. 

97 

55 

42. 74 

34. 61 

42. 59 

3 4. 80 

42. 44 

34. 98 

42, 29 

35. 

17 

GO 

46. 63 

37. 76 

46. 46 

37. 96 

46. 30 

38. 16 

46. 13 

38. 

37 

65 

50. 51 

40. 91 

50. 34 

41. 13 

50. 16 

41. 35 

49. 97 

41. 

56 

70 

54. 40 

44. 05 

54. 21 

44. 29 

54. 01 

44. 53 

53. 82 

44. 

76 

75 

58. 29 

47. 20 

58. 08 

47. 45 

57. 87 

47. 71 

57. 66 

47. 

96 

80 

62. 17 

50. 35 

61. 95 

50. 62 

61. 73 

50. 89 

61. 51 

51. 

16 

85 

66. 06 

53. 49 

65. 82 

53. 78 

65. 59 

54. 07 

65. 35 

54. 

35 

90 

69. 94 

56. 64 

69. 70 

56. 94 

69. 45 

57. 25 

69. 20 

57. 

55 

95 

73. 83 

59. 79 

73. 57 

60. 11 

73. 30 

60. 43 

73. 04 

60. 

75 

100 

77. 71 

62. 93 

77. 44 

63. 27 

77. 16 

63. 61 

76. 88 

63. 

94 

4 

o 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lilti 

5 

00 

5 

51 Deo. 

50# 

Df.«. 

60# 

Deo. 

60# 

Deo 



































































— 



TRAVERSE 

TABLE. 


111 

tj 

o 

c 

40 

Deo. 

40^ 

Deo. 

40K 

Deo. 

v*# 

o 

t 

i 

1 

Deo. 

5 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 77 

0. 64 

0. 76 

0. 65 

0. 76 

0. 65 

0. 76 

0. 65 

2 

1. 53 

1. 29 

1. 53 

1. 29 

1. 52 

1. 30 

1. 52 

1. 31 

3 

2. 30 

1. 93 

2. 29 

1. 94 

2. 28 

1. 95 

2. 27 

1. 96 

4 

3. 06 

2. 57 

3. 05 

2. 58 

3. 04 

2. 60 

3. 03 

2. 61 

5 

3. 83 

3. 21 

3. 82 

3. 23 

3. 80 

3. 25 

3. 79 

3. 26 

6 

4. 60 

3. 86 

4. 58 

3. 88 

4. 56 

3. 90 

4. 55 

3. 92' 

7 

5. 36 

4. 50 

5. 31 

4. 52 

5. 32 

4. 55 

5. 30 

4. 57 

8 

6. 13 

5. 14 

6. 11 

5. 17 

6. 08 

5. 20 

6. 06 

5. 22 

9 

6. 89 

5. 79 

6. 87 

5. 82 

6 81 

5. 84 

6. 82 

5. 87 

10 

7. 66 

6. 43 

7. 63 

6. 46 

7. 60 

6. 49 

7. 58 

6. 53 

11 

8. 43 

7. 07 

8. 10 

7. 11 

8. 36 

7. 14 

8. 33 

7. 18 

12 

9. 19 

7. 71 

9. 16 

7. 75 

9. 12 

7. 79 

9. 09 

7. 83 

13 

9. 96 

8. 36 

9. 92 

8. 40 

9. 89 

8. 41 

9. 85 

8. 49 

14 

10. 72 

9. 00 

10. 69 

9. 05 

10. 65 

9. 09 

10. 61 

9. 14 

15 

11. 49 

9. 64 

11. 45 

9. 69 

11. 41 

9. 74 

11. 86 

9. 79 

16 

12. 26 

10. 28 

12. 21 

10. 31 

12. I V 

10. 89 

12. 12 

10. 41 

17 

13. 02 

10. 93 

12. 97 

10. 98 

12. 93 

11. 04 

12. 88 

11. 10 ! 

18 

13. 79 

11. 57 

13. 74 

11 63 

13. 69 

11. 69 

13. 64 

11. 75 

19 

14. 55 

12. 21 

14. 50 

12. 28 

14. 45 

12. 34 

14. 39 

12. 40 

20 

15. 32 

12. 86 

15. 26 

12. 92 

15. 21 

12. 99 

15. 15 

13. 06 

21 

16. 09 

13. 50 

16. 03 

13. 57 

15. 97 

13. 64 

15. 91 

13. 71 

22 

16. 85 

14. 14 

16. 79 

14. 21 

16 73 

14. 29 

16. 67 

14. 36 ! 

23 

17. 62 

14. 78 

17. 55 

14. 86 

17. 49 

14. 94 

17. 42 

15. 01 1 

24 

18. 39 

15. 43 

18. 32 

15. 51 

18. 25 

15. 59 

18. 18 

15. 67 1 

25 

19. 15 

16. 07 

19. 08 

16. 15 

19. 01 

16. 24 

18. 94 

16. 32 

26 

19. 92 

16. 71 

19. 84 

16. 80 

19. 77 

16. 89 

19. 70 

16. 97 

27 

20. 63 

17. 36 

20. 61 

17. 45 

20. 53 

17. 54 

20. 45 

17. 62 

28 

21. 45 

18. 00 

21. 37 

18. 09 

2!. 29 

18. 18 

21. 21 

18. 23 

29 

22. 22 

18. 64 

22. 13 

18. 74 

22. 05 

18. 83 

21. 97 

18. 93 

30 

22. 98 

19. 28 

22. 90 

19. 38 

22. 81 

19. 48 

22. 73 

19. 58 

35 

26. 81 

22. 50 

26. 71 

22. 61 

26. 61 

22. 73 

26. 51 

22. 85 

40 

30. 64 

25. 71 

30. 53 

25. 84 

30. 42 

25. 98 

30. 30 

26. 11 

45 

34. 47 

28. 93 

34. 35 

29. 08 

34. 22 

29. 23 

34. 09 

29. 37 

50 

38. 30 

32. 14 

38. 16 

32. 31 

38. 02 

32. 47 

37. 88 

32. 64 

55 

42. 13 

35. 35 

41. 98 

35. 54 

41. 82 

35. 72 

41. 67 

35. 90 

60 

45. 96 

38. 57 

45. 79 

38. 77 

45. 62 

38. 97 

45. 45 

39. 17 

65 

49. 79 

41. 78 

49. 61 

42. 00 

49. 43 

42. 21 

49. 24 

42. 43 

70 

53. 62 

45. 00 

53. 43 

45. 23 

53. 23 

45. 46 

53. 03 

45. 69 

75 

57. 45 

48. 21 

57. 21 

48. 46 

57. 03 

48. 71 

56. 82 

48. 96 

80 

61. 28 

51. 42 

61. 06 

51. 69 

60. 83 

51. 96 

60. 61 

52. 22 

85 

65. 11 

54. 64 

61. 87 

54. 92 

64. 63 

55. 20 

64. 39 

55. 48 

90 

68. 94 

57. 85 

68. 69 

58 15 

68. 44 

58. 45 

68. 18 

58. 75 

95 

72. 77 

61. 06 

72. 51 

61. 38 

72. 24 

61. 70 

71. 97 

62. 01 

100 

76. 60 

64. 28 

76. 32 

64. 61 

76. 04 

64. 94 

75. 76 

65. 28 

V 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Sj 

«-> 

5 

50 Deo. 

49^ Deo. j 

49)4 Deo. 

4 9>4 

Deo. 








































































112 TRAVERSE TABLE. 


V 

v 

p 

7i 

41 Deo. 

41* 

Deo. 

41* 

Deg. 


41*7 

Deo. 

r» 


Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0. 75 

0. 

66 

0. 75 

0. 66 

0. 75 

0. 

66 

0. 75 

0. 67 

2 

1. 51 

1. 

31 

1. 50 

1. 32 

1. 50 

1. 

33 

1. 49 

1. 33 

3 

2. 26 

1. 

97 

2. 26 

1. 98 

2. 25 

1. 

99 

2. 24 

2. 00 

4 

3. 02 

2. 

62 

3. 01 

2. 64 

3. 00 

2. 

65 

2. 98 

2. 66 

6 

3. 77 

3. 

28 

3. 76 

3. 30 

3. 74 

3. 

31 

3. 73 

3. 33 

6 

4. 53 

3. 

94 

4. 51 

3. 96 

4. 49 

3. 

98 

4. 48 

4. 00 

7 

5. 28 

4. 

59 

5. 26 

4. 62 

5. 21 

4. 

64 

5. 22 

4. 66 

8 

6. 04 

5. 

25 

6. 01 

5. 27 

5. 99 

5. 

30 

5. 97 

5. 33 

9 

6. 79 

5. 

90 

6. 77 

5. 93 

6. 74 

5. 

96 

6. 71 

5. 99 

10 

7. 55 

6. 

56 

7. 52 

6. 59 

7. 49 

6. 

63 

7. 46 

6. 66 

11 

8. 30 

7. 

22 

8. 27 

7. 25 

8. 24 

7. 

29 

8. 21 

7. 32 

12 

9. 06 

7. 

87 

9. 02 

7. 91 

8. 99 

7. 

95 

8. 95 

7. 99 

13 

9. 81 

8. 

53 

9. 77 

8 57 

9. 74 

8 

61 

9. 70 

8. 66 ; 

14 

10. 57 

9. 

18 

10. 53 

9. 23 

10. 49 

9 

28 

10. 44 

9. 82 

15 

11. 32 

9. 

84 

11. 28 

9. 80 

11. 23 

9. 

94 

11. 19 

9. 99 

16 

12. 08 

10 

50 

12. 03 

10. 55 

11. 98 

10. 

6u 

11. 94 

10. 65 

17 

12. 83 

11. 

15 

12. 78 

11. 21 

12. 73 

11. 

26 

12. 68 

11. 32 

18 

13. 58 

11. 

81 

13. 53 

11. 87 

13. 48 

11. 

93 

13. 43 

11. 99 

19 

14 34 

12. 

47 

14. 28 

12. 53 

14. 23 

12. 

59 

14. 18 

12. 65 

20 

15. 09 

13. 

12 

15. 04 

13. 19 

14. 98 

13. 

25 

14. 92 

13. 32 

21 

15. 85 

13. 

78 

15. 79 

13. 85 

15. 73 

13. 

91 

15. 67 

13. 98 

22 

16. 60 

14. 

43 

16. 54 

14. 51 

16. 48 

14. 

58 

16. 41 

14. 65 

23 

17. 36 

15. 

09 

17. 29 

15. 16 

17. 23 

15. 

24 

17. 16 

15. 32 

24 

18. 11 

15. 

75 

18. 04 

15. 82 

17. 97 

15. 

00 

17. 91 

15. 98 1 

25 

18. 87 

16. 

40 

18. 80 

16. 48 

18. 72 

16. 

57 

18. 65 

16. 65 

26 

19. 62 

17. 

06 

19. 55 

17. 14 

19. 47 

17. 

23 

19. 40 

17. 31 

27 

20. 38 

17. 

71 

20. 30 

17. 80 

20. 22 

17. 

89 

20. 14 

17. 98 

28 

21. 13 

18. 

37 

21. 05 

18. 46 

20. 97 

18. 

55 

20. 89 

18. 64 

29 

21. 89 

19. 

03 

21. 80 

19. 12 

21. 72 

19. 

22 

21. 64 

19. 31 

30 

22. 64 

19. 

68 

22. 56 

19. 78 

22. 47 

19. 

88 

2 \ 38 

19. 98 

35 

26. 41 

22. 

96 

26. 31 

23. 08 

26. 21 

23. 

19 

26. 11 

23. 31 

40 

30. 19 

26. 

24 

30. 07 

26. 37 

29. 96 

26. 

50 

29. 84 

26. 64 

45 

33. 96 

29. 

52 

33. 83 

29. 67 

33. 70 

29. 

82 

33. 57 

29. 97 

50 

37. 74 

32. 

8) 

37. 59 

32. 97 

37. 45 

33. 

13 

37. 30 

33. 29 

55 

41 51 

36. 

08 

41. 35 

36 26 

41. 19 

36. 

44 

41. 03 

36. 62 

60 

45. 28 

39. 

36 

45. 11 

39. 56 

44. 94 

39. 

76 

44. 76 

39. 95 

65 

49. 06 

42. 

64 

48. 87 

42. 86 

48. 68 

43. 

07 

48. 49 

43. 28 

70 

52. 83 

45. 

92 

52. 63 

46. 15 

52. 43 

46. 

88 

52. 22 

46. 61 

75 

56. 60 

49. 

20 

56. 39 

49. 45 

56. 17 

49. 

70 

55. 95 

4 9. 94 

80 

60. 38 

52. 

48 

60. 15 

52. 75 

59. 92 

53. 

01 

59. 68 

53. 27 

85 

64. 15 

55. 

76 

63. 91 

56. 04 

63. 66 

56. 

32 

63. 41 

56. 60 

90 

67. 92 

59 

05 

67. 67 

59. 31 

67. 41 

59. 

64 

67. 15 

59. 93 

95 

71. 70 

62. 

33 

71. 43 

62. 64 

71. 15 

62. 

95 

70. 88 

63. 26 

100 

75. 47 

65. 

61 

75. 18 

65. 93 

74. 90 

66. 

26 

74, 61 

66 59 

« 

o 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

03 

v> 

5 

49 Deo. 

mi 

Deo. 

48* Deo. 

48* 

Deg. 







































































TRAVERSE 

TABLE. 




""" 

113 i 

§ 

rt 

w 

5 

42 Deg. 

4-% Df.o, 


42% Deg. 


—-- j 

42% Deo. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. ^ 

i 

0. 74 

0. 

67 

0 

74 

0. 

07 

0. 

74 

0. 

68 

0. 

73 

0. 

68 

2 

1. 49 

1. 

34 

1. 

48 

1. 

34 

1. 

47 

1. 

35 

1. 

47 

1. 

36 

3 

2. 23 

2. 

01 

2. 

22 

2. 

02 

2. 

21 

2. 

03 

2. 

2 > 

2. 

04 

4 

2. 97 

2. 

68 

2. 

96 

2. 

69 

2. 

95 

2. 

70 

2 

91 

2. 

72 

5 

3. 72 

3. 

35 

3. 

70 

3. 

36 

3. 

69 

3. 

3S 

3. 

67 

3 

39 

6 

4. 46 

4. 

01 

4. 

44 

4. 

03 

4. 

42 

4. 

05 

4 

41 

4. 

07 

7 

5. 20 

4. 

68 

5. 

18 

4. 

71 

5. 

16 

4. 

73 

5. 

14 

4. 

75 

. 8 

5. 95 

5. 

35 

5. 

92 

5. 

38 

5. 

90 

5. 

40 

5. 

87 

5. 

43 

9 

6. 69 

6. 

02 

6. 

66 

6. 

05 

6. 

61 

6. 

08 

6. 

61 

6. 

11 i 

10 

7. 43 

6. 

69 

7. 

40 

6. 

72 

7. 

37 

6. 

76 

7. 

34 

6. 

79 

11 

8. 17 

7. 

36 

8. 

14 

7. 

40 

8. 

11 

7. 

43 

8. 

08 

7. 

47 j 

12 

8. 92 

8. 

03 

8. 

88 

8. 

07 

8. 

85 

8. 

11 

8. 

81 

8. 

15 ] 

13 

9. 66 

8. 

70 

9. 

62 

8. 

74 

9. 

5S 

8. 

78 

9. 

55 

8. 

82 | 

14 

10. 40 

9. 

37 

10. 

36 

9. 

41 

10. 

33 

9. 

46 

10. 

28 

9. 

50 | 

15 

11. 15 

10. 

04 

11. 

10 

10. 

09 

11. 

06 

10. 

13 

11. 

01 

10. 

18 

16 

11. 89 

10. 

71 

11, 

84 

10. 

76 

11. 

80 

10. 

81 

11. 

75 

10. 

86 ! 

17 

12. 63 

11. 

3S 

12. 

58 

11. 

43 

12. 

53 

11. 

48 

12. 

48 

11. 

54 j 

18 

13. 38 

12. 

04 

13. 

32 

12. 

10 

13. 

27 

12. 

16 

13. 

22 

12. 

22 j 

1 19 

14. 12 

12. 

71 

14. 

06 

12. 

77 

14. 

01 

12. 

84 

13. 

95 

12. 

90 ! 

i 20 

14. 86 

13. 

38 

14. 

SO 

13. 

45 

14. 

75 

13. 

51 

14. 

69 

13. 

58 j 

! 21 

15. 61 

14. 

05 

15. 

54 

14. 

12 

15. 

48 

14. 

19 

15. 

42 

14. 

25 ! 

| 22 

16. 35 

14. 

72 

16. 

28 

14. 

79 

16. 

22 

14. 

86 

16. 

16 

14. 

93 j 

23 

17. 09 

15. 

39 

17. 

02 

15. 

46 

16. 

96 

15. 

54 

16. 

89 

15. 

61 

24 

17. 81 

16. 

06 

17. 

77 

16. 

14 

17. 

69 

16. 

21 

17. 

62 

16. 

29 

25 

18. 5S 

16. 

73 

18. 

51 

16. 

81 

18. 

48 

16. 

89 

18. 

36 

16. 

97 

26 

19. 32 

17. 

40 

19. 

25 

17. 

48 

19. 

17 

17. 

57 

19. 

09 

17. 

65 

| 27 

20. 06 

18. 

07 

19. 

99 

18. 

15 

19. 

91 

18. 

24 

19. 

83 

18. 

3 • 

28 

20. 81 

18. 

74 

20. 

73 

18. 

83 

20. 

64 

18. 

92 

20. 

56 

19. 

01 

29 

21. 55 

19 

40 

21. 

47 

19. 

50 

21. 

38 

19. 

59 

21. 

30 

19. 

69 

| 30 

22. 29 

20. 

07 

22. 

21 

20. 

17 

22. 

12 

20. 

27 

22. 

03 

20. 

36 ! 

[ 35 

26. 01 

23. 

42 

25. 

91 

23. 

53 

25. 

80 

23. 

65 

25. 

70 

23. 

76 1 

40 

29. 73 

26. 

77 

29. 

61 

26. 

89 

29. 

49 

27. 

02 

29. 

37 

27. 

15 

45 

33. 44 

30. 

11 

33. 

31 

30. 

26 

33. 

18 

30. 

40 

33. 

04 

30. 

55 

50 

37. 16 

33. 

46 

37. 

oi 

33 

62 

36. 

86 

33. 

78 

86. 

72 

33. 

94 

55 

40. 87 

36. 

80 

40. 

71 

36. 

98 

40. 

55 

37. 

16 

40. 

39 

37. 

33 

60 

44. 59 

40. 

15 

44. 

41 

40. 

34 

44. 

24 

40. 

54 

44. 

06 

40. 

'<3 

65 

48. 30 

43. 

49 

48. 

11 

43. 

70 

47. 

92 

43. 

91 

47. 

73 

44. 

12 , 

7C 

52. 02 

46. 

84 

51. 

82 

47. 

07 

51. 

61 

47. 

29 

51. 

40 

47. 

52 

75 

55. 74 

50. 

18 

55. 

52 

50. 

43 

55. 

30 

50. 

67 

55. 

07 

50. 

91 

80 

59. 45 

53. 

53 

59. 

22 

53. 

79 

58. 

98 

54. 

05 

58. 

75 

54. 

30 

85 

63. 17 

56. 

88 

62. 

92 

57. 

15 

62. 

67 

57. 

43 

62. 

42 

57. 

70 

90 

66. 88 

60. 

22 

66. 

63 

60. 

51 

66. 

35 

60. 

80 

66. 

09 

61. 

09 

95 

70. 60 

63. 

57 

70. 

32 

63. 

87 

70. 

04 

64. 

18 

69. 

76 

64. 

49 

100 

74. 31 

66. 

61 

74. 

02 

67. 

24 

73. 

73 

67. 

56 

73. 

43 

67. 

88 

6 

o 

P 

'A 

CO 

5 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

48 Deo. 

47% 

Deo. 


47% Deo. 

47% Deo. 
















































































114 


TRAVERSE 

TABLE. 



— " 

6 

o 

j 

* 

43 Deo. 

43% Deg. 

4VA 

Deg. 


4 % 

Deo. 

» 

’ 5 

Lat. 

Dep. 

Lat 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

i 

0. 73 

0. 63 

0. 73 

0. 69 

0. 73 

0. 

69 

0. 72 

0. 69 

2 

1. 46 

1. 38 

1. 46 

1. 37 

1. 45 

1. 

38 

1. 44 

1. 38 

3 

2. 19 

2. 05 

2. 19 

2. 06 

2. 18 

2. 

07 

2. 17 

2. 07 

4 

2. 93 

2. 73 

2. 91 

2. 74 

2. 90 

2. 

75 i 

2. 89 

2. 77 

5 

3. 66 

3. 41 

3. 64 

•3. 43 

3. 63 

3. 

44 

3. 61 

3. 46 

6 

4. 39 

4. 09 

4. 37 

4. 11 

4. 35 

4. 

13 

4. 33 

4. 15 

7 

5. 12 

4. 77 

5. 10 

4. 80 

5. OS 

4. 

82 

5. 06 

4. 84 

8 

5. 85 

5. 46 

5. 83 

5. 48 

5. 80 

5. 

51 

5. 78 

5. 53 

9 

6. 58 

6. 14 

6. 56 

6. 17 

6. 53 

6. 

20 

6. 50 

6. 22 

10 

7. 31 

6. 82 

7. 28 

6. 85 

7. 25 

6. 

88 

7. 22 

6. 92 

11 

8. 04 

7. 50 

8. 01 

7. 54 

7. 98 

7. 

57 

7. 95 

7. 61 

12 

8. 78 

8. 18 

8. 74 

8. 22 

8. 70 

8. 

26 

8. 67 

8. 30 

13 

9. 51 

8. 87 

9. 47 

8. 91 

9. 43 

8. 

95 

9. 39 

8. 99 

14 

10. 21 

9. 55 

10. 20 

9. 59 

10. 16 

9. 

64 

10. 11 

9. 68 

15 

10. 97 

10. 23 

10. 93 

10. 28 

10. 88 

10. 

33 

10. 84 

10. 37 

16 

11. 70 

10. 91 

11. 65 

10. 96 

11. 61 

11. 

01 

11. 56 

11. 06 

17 

12. 43 

11. 59 

12. 38 

11. 65 

12. 33 

11. 

70 

12. 28 

11. 76 

18 

13. 16 

12. 28 

13. 11 

12. 33 

13. 06 

12. 

39 

13. 00 

12. 45 

19 

13. 90 

12. 96 

13. 84 

13. 02 

13. 78 

13. 

08 

13. 72 

13. 14 

20 

11. 63 

13. 64 

14 57 

13. 70 

14. 51 

13. 

77 

14. 45 

13. 83 

21 

15. 36 

14. 32 

15. 30 

14. 39 

15. 23 

14. 

46 

15. 17 

14. 52 

22 

16. 09 

15. 00 

16. 02 

15. 07 

15. 96 

15. 

14 

15 89 

15. 21 

23 

16. 82 

15. 69 

16. 75 

15. 76 

16 63 

15. 

83 

16. 61 

15. 90 

21 

17. 55 

16. 37 

17. 4S 

16. 41- 

17. 41 

16. 

52 

17. 34 

16. 60 

25 

13. 28 

17. 05 

18. 21 

17. 13 

18. 13 

17. 

21 

18. 06 

17. 29 

26 

19. 02 

17. 73 

18. 94 

17. 81 

18. 86 

17. 

90 

18. 78 

17. 98 

27 

19. 75 

18. 41 

19. 67 

18. 50 

19. 59 

18. 

59 

19. 50 

18. 67 

28 

20. 48 

19. 10 

20. 39 

19. 19 

2<». 31 

19. 

27 

20. 23 

19. 36 

29 

21. 21 

19. 73 

21. 12 

19. 87 

21. 04 

19. 

96 

20. 95 

20. 05 

30 

21. 94 

20. 46 

21 85 

20. 56 

21. 76 

20. 

65 

21. 67 

20. 75 

35 

25. 60 

23. 87 

25. 49 

23. 98 

25. 39 

24. 

09 

25. 28 

24. 20 

40 

2). 25 

27. 23 

29. 13 

27. 41 

29. 01 

27. 

53 

28. 89 

27. 66 

45 

32. 91 

30. 69 

32. 78 

30. 83 

32. 64 

30. 

98 

32. 51 

31. 12 

50 

36. 57 

34. 10 

36. 42 

34. 26 

36 27 

34. 

42 

36. 12 

34. 58 

55 

40. 2 1 

37. 51 

40. 06 

37. 69 

39. 90 

37. 

86 

39. 73 

38. 03 

60 

43. 88 

40. 92 

43. 70 

41. 11 

43. 52 

41. 

30 

43. 31 

41. 49 

65 

47. 54 

44. 33 

47. 34 

41. 54 

47. 15 

44. 

74 

46. 95 

44. 95 

1 70 

51. 19 

47. 74 

50. 99 

47. 96 

50. 78 

48. 

18 

50 57 

; 48. 41 

75 

54. 85 

51. 15 

54. 63 

51. 39 

54. 40 

51. 

63 

54. 18 

51. 86 

80 

53. 51 

54. 56 

58. 27 

54. 81 

58. 03 

55. 

07 

57. 79 

55. 32 

85 

62. 17 

57. 97 

61. 91 

58. 24 

61. 66 

58. 

51 

61. 40 

58. 78 

90 

65. 82 

61. 38 

65. 55 

61. 67 

65. 28 

61. 

95 

65. 01 

62. 24 

95 

69. 48 

64. 79 

69. 20 

65. 09 

68. 91 

65 

39 

68. 62 

65. 69 

100 

73. 14 

68. 20 

72. 84 

68. 52 

72. 54 

68 

84 

72. 24 

69. 15 

6 

o 

a 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

L<vt. 

rt 

cc 

5 

47 Deo. 

46 % 

Deo. 

46 % 

Deo 

• 

46 % 

De®. 










































































TRAVERSE TABLE. 115 


i, -»*.- 

6 

x ^ 

r: 

es 

44 Di;g. 

41% Deg. 

44% Deg. 

44% 

Deg. 

45 Deg. 

i n 

5 

I.at. 

l>ep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dtp. 

Lat. 

Dtp. 

1 

0.72 

0.69 

0.72 

0.70 

0.71 

0.70 

0.71 

0.71 

0.71 

0.71 

2 

1.44 

1.39 

1.43 

1.40 

1.43 

1.40 

1.42 

1.41 

1.41 

1.41 

3 

2 16 

2.08 

2 15 

2.09 

2.14 

2.10 

2,13 

2.11 

2.12 

2.12 

4 

2.88 

2.78 

2.87 

2.79 

2.85 

2.80 

2.84 

2 82 

2.83 

2.83 

5 

3.60 

3,47 

3.58 

3.49 

3.57 

3.50 

3.55 

3.52 

3.54 

3.54 I 

6 

4 32 

4.17 

4.30 

4.19 

4.28 

4.21 

4.26 

4.22 

4.24 

4.24 

7 

5.04 

4 86 

5.01 

4.88 

4.99 

4.91 

4.97 

4.93 

4.95 

4.95 

8 

5.75 

5.56 

5.73 

5.58 

5.71 

5.61 

5.68 

5.63 

5.66 

5.66 

9 

6.47 

6.25 

6.45 

6.28 

6.42 

6.31 

6.39 

6.34 

6.36 

6.36 

10 

7.19 

6 95 

*16 

6.98 

7.13 

7.01 

7.10 

7.04 

7.07 

7.07 

11 

7.91 

7.64 

7.88 

7.68 

7.86 

7.71 

7.81 

7.74 

7.78 

7.78 

12 

8.63 

8.34 

8.60 

8.37 

8.56 

8 41 

8.52 

8.45 

8.49 

8.49 

13 

9.35 

9 03 

9.31 

9.07 

9.27 

9.11 

9.23 

9.15 

9.19 

9.19 

14 

10.07 

9.73 

10 03 

9.77 

9.99 

9.81 

9.94 

9.86 

1 9.00 

9 90 

15 

10.79 

10.42 

10.74 

10.74 

10.70 

10.51 

10.65 

10.56 

10.61 

10.61 

16 

11 51 

11.11 

11.46 

11 16 

11.41 

11.21 

11.36 

11.26 

11.31 

11.31 

17 

12.23 

11.81 

12.18 

11 86 

12 13 

11.92 

12.07 

11.97 

12.02 

12.02 

18 

12.95 

12.50 

12.89 

12.56 

12.84 

12.62 

12.78 

12,67 

12.73 

12.73 

19 

13.67 

13.20 

13.61 

13.26 

13.55 

13.32 

13.49 

13.38 

13.43 

13.43 

20 

14.39 

13.89 

14.33 

13 96 

14.26 

14.02 

14.20 

14.08 

14.14 

14.14 

21 

15.11 

14.59 

15.04 

14.65 

14.98 

14.72 

14.91 

14.78 

14 85 

14.85 

22 

15.83 

15.28 

15 76 

15.35 

15.69 

15.42 

15.62 

15.49 

j 15.56 

15.56 

23 

16.54 

15.98 

16.47 

16.05 

16.40 

16.12 

16.33 

16.19 

16.26 

16.26 

24 

17.26 

16.67 

17.19 

16.75 

17.12 

16.82 

17.04 

16.90 

! 16.97 

16.97 

25 

17.98 

17.37 

17.91 

17.44 

17.83 

17.52 

17.75 

17.60 

! 17.68 

17.68 

26 

18.70 

18.06 

18.62 

18.14 

18.54 

18.22 

18.46 

18,30 

18.38 

18 38 

27 

19.42 

18.76 

19.34 

18.84 

19.26 

18.92 

19.17 

19.01 

19.09 

19.09 

28 20.14 

19.45 

20.06 

19.54 

19.97 

19.63 

19.89 

19,71 

19.80 

19.80 

29 

20.86 

20.15 

20.77 

20.24 

20.68 

20.33 

20.60 

20.42 

20.51 

20.51 

30 

21.58 

20.84 

21.49 

20.93 

21.40 

21.03 

21.31 

21.12 

21.21 

21.21 

35 

25.18 

24.31 

25 07 

24.42 

24.96 

24.53 

24.86 

24.64 

24.75 

24.75 

40 

28.77 

27.79 

28.65 

27.91 

28.53 

28.04 

28.41 

23.16 

28.28 

28.28 

4') 

32.37 

31.26 

32.23 

31.40 

32.10 

31.54 

31.96 

31.68 

31.82 

31.82 

50 

35.97 

34.73 

35.82 

31.89 

35.66 

35.05 

35.51 

35.20 

35.36 

35.36 

55 

39 56 

38.21 

39.4(' 

38.38 

39.23 

38.55 

39.06 

38.72 

38.89 

38.89 

60 

43.16 

41.68 

42.98 

41.87 

42.79 

42.05 

42 61 

42.24 

42.43 

42.43 

65 

46.76 

45.15 

46.56 

45.36 

46.36 

45.56 

46.16 

45.76 

45.9645.96 

7050.35:48.63 

50.14 

48.85 

49.93 

49.06 

49.71 

49.28 

49.50 

49 50 

75 

53.95 

52.10 

53.72 

52.33 

53.49 

52.57 

53.26 

52.80 

53.03 

53.03 

80 

57.55 

55.57 

57.30 

55.82 

57.06 

56.07 

56.81 

56.32 

56.57 

56.57 

85 

61.14 

59.05 

60.89 

59.31 

60.63 

59.58 

60.37 

59.84 

60.10 

60.10 

90 

64.74 

62.52 

64.47 

62.80 

64.19 

63.08 

63.92 

63.36 

63.64 

63.64 

95 

68.34 

65.99 

68.05 

66.29 

67.76 

66.59 

67.47 

66.88 

67.18 

67.18 

100 

71 93 

69.47 

71.63 

69.78 

71.33 

70.09 

71.02 

70 40 

70.71 

70.71 

© 

© 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

i Dep. 

Lat. 

e 

08 

f <-* 

5 

46 Deo. 

45% Deo. 

45% Deg. 

45% Deo. 

45 Deg. 

i 

























































































Meridlai al Parts. TABLE IV. 


# 

if 

1 ° 

2° 

8° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

i no 

J O 

14 e 

15* 

0 

0 

60 

120 

180 

240 

800 

361 

421 

482 

542 

603 

064 

725 

787 

848 

910 

1 

1 

61 

121 

181 

241 

101 

362 

422 

488 

543 

604 

665 

726 

788 

850 

911 

2 

2 

62 

122 

182 

242 

302 

363 

423 

484 

544 

605 

666 

727 

789 

851 

913 

S 

3 

63 

128 

183 

243 

303 

364 

424 

485 

545 

606 

667 

728 

790 

852 

914 

4 

4 

64 

124 

184 

244 

304 

365 

425 

486 

546 

6t)7 

668 

729 

791 

853 

915 

5 

5 

65 

] 25 

185 

245 

305 

366 

426 

487 

547 

608 

669 

730 

792 

854 

916 

6 

6 

66 

126 

186 

246 

806 

367 

427 

488 

548 

609 

670 

731 

793 

855 

917 

7 

7 

67 

127 

187 

247 

307 

368 

428 

489 

549 

61d 

671 

782 

794 

856 

918 

8 

8 

68 

128 

188 

248 

308 

369 

429 

490 

550 

611 

672 

734 

795 

857 

919 

9 

9 

69 

129 

189 

249 

309 

370 

430 

491 

551 

612 

673 

735 

796 

858 

920 

10 

10 

70 

130 

190 

250 

310 

371 

431 

492 

552 

613 

674 

736 

797 

859 

921 

11 

11 

71 

131 

191 

251 

311 

372 

432 

493 

553 

614 

675 

787 

798 

860 

922 

12 

12 

72 

132 

192 

252 

312 

373 

433 

494 

554 

615 

676 

738 

799 

861 

923 

18 

13 

73 

133 

193 

253 

313 

374 

434 

49. 

555 

616 

677 

739 

800 

862 

924 

14 

14 

74 

134 

194 

254 

314 

375 

435 

496 

556 

617 

678 

740 

801 

863 

925 

15 

15 

75 

135 

195 

255 

315 

876 

436 

497 

557 

61» 

679 

741 

802 

864 

926 

10 

10 

76 

136 

196 

256 

316 

377 

437 

498 

558 

619 

681) 

742 

808 

865 

927 

17 

17 

77 

137 

197 

257 

317 

378 

438 

499 

559 

620 

681 

743 

804 

866 

9^8 

18 

18 

78 

138 

198 

258 

318 

379 

439 

500 

560 

621 

682 

744 

805 

867 

929 

19 

19 

79 

189 

199 

259 

319 

380 

440 

501 

561 

622 

683 

745 

806 

868 

930 

20 

20 

80 

140 

200 

260 

320 

381 

441 

502 

562 

623 

684 

746 

807 

869 

931 

21 

21 

81 

141 

201 

261 

321 

382 

442 

503 

564 

624 

685 

747 

808 

870 

932 

22 

22 

82 

142 

202 

262 

322 

383 

443 

504 

565 

625 

687 

748 

809 

871 

933 

28 

23 

83 

143 

208 

263 

323 

384 

444 

505 

566 

626 

68 

749 

810 

872 

934 

24 

24 

84 

144 

204 

264 

324 

385 

445 

5o6 

567 

627 

689 

750 

811 

873 

935 

25 

25 

85 

1 i 5 

205 

265 

325 

386 

446 

507 

568 

628 

690 

751 

812 

874 

936 

20 

26 

80 

146 

206 

266 

326 

387 

447 

508 

569 

629 

691 

752 

818 

875 

937 

27 

27 

S7 

147 

207 

267 

327 

388 

448 

509 

570 

631 

692 

753 

815 

876 

988 

2b 

28 

88 

148 

208 

268 

328 

389 

449 

510 

571 

632 

693 

754 

816 

877 

989 

29 

29 

89 

149 

209 

269 

330 

390 

450 

511 

572 

633 

694 

755 

817 

878 

941 

80 

30 

90 

150 

210 

270 

331 

391 

451 

512 

573 

634 

695 

756 

818 

879 

942 

31 

31 

91 

151 

211 

271 

332 

392 

452 

513 

574 

635 

696 

757 

819 

880 

943 

32 

32 

92 

152 

212 

272 

333 

393 

453 

514 

575 

636 

697 

758 

820 

882 

944 

38 

S3 

93 

153 

213 

273 

384 

394 

454 

515 

576 

637 

698 

759 

821 

883 

945 

34 

34 

94 

154 

214 

274 

335 

395 

455 

516 

577 

638 

699 

7*0 

822 

884 

946 

35 

35 

95 

155 

215 

275 

336 

396 

456 

517 

578 

639 

700 

761 

823 

885 

947 

86 

30 

96 

156 

216 

276 

337 

397 

457 

518 

579 

640 

7ol 

762 

824 

886 

948 

37 

87 

97 

157 

217 

277 

838 

398 

458 

519 

580 

641 

702 

763 

825 

887 

949 

38 

38 

98 

158 

218 

278 

339 

399 

459 

520 

58l 

642 

703 

764 

826 

888 

950 

39 

39 

99 

159 

219 

279 

340 

400 

460 

521 

582 

643 

704 

765 

827 

839 

951 

40 

40 

100 

160 

220 

2S0 

341 

401 

461 

522 

f 83 

644 

705 

766 

828 

890 

952 

41 

41 

101 

161 

221 

281 

342 

402 

462 

523 

584 

645 

706 

767 

829 

891 

953 

42 

42 

102 

162 

222 

282 

343 

403 

463 

524 

585 

646 

707 

768 

830 

892 

954 

43 

48 

103 

103 

223 

283 

344 

404 

464 

525 

586 

647 

708 

769 

831 

893 

955 

44 

44 

104 

164 

224 

284 

845 

405 

465 

526 

587 

648 

709 

770 

832 

894 

956 

45 

45 

105 

165 

225 

285 

346 

406 

466 

527 

588 

649 

710 

771 

833 

895 

957 

40 

46 

106 

166 

226 

286 

347 

407 

467 

528 

589 

650 

711 

772 

834 

896 

958 

47 

47 

107 

167 

227 

287 

348 

408 

468 

529 

590 

651 

712 

773 

835 

897 

959 

48 

48 

108 

168 

228 

288 

349 

4<>9 

469 

530 

591 

652 

718 

774 

836 

898 

960 

49 

49 

109 

169 

229 

289 

350 

410 

470 

531 

592 

$53 

714 

775 

837 

899 

961 

50 

50 

no 

170 

230 

200 

851 

411 

471 

532 

593 

654 

715 

777 

838 

900 

962 

51 

51 

111 

171 

231 

291 

352 

412 

472 

533 

594 

655 

716 

778 

839 

901 

963 

52 

52 

112 

172 

232 

292 

353 

413 

473 

534 

595 

656 

717 

779 

840 

902 

964 

63 

53 

113 

173 

233 

293 

354 

414 

474 

535 

596 

657 

718 

780 

841 

903 

965 

64 

54 

114 

174 

234 

294 

355 

415 

476 

536 

597 

658 

719 

781 

842 

904 

966 

55 

55 

115 

175 

235 

295 

356 

416 

477 

537 

598 

659 

720 

782 

843 

905 

968 

56 

56 

116 

176 

286 

296 

357 

417 

478 

538 

599 

600 

721 

783 

844 

906 

969 

57 

57 

117 

177 

287 

297 

358 

418 

479 

539 

600 

661 

722 

784 

845 

907 

970 

68 

53 

118 

178 

23b 

298 

359 

419 

480 

540 

601 

602 

723 

785 

846 

908 

971 

59 

59 

119 

179 

239 

299 


420 

1 481 

541 

602 

668 

724 

786 

847 

909 

972 



















































TABLE IV. 



Meridianal Parts. 





117 

' • 

16° 

\r 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

0 

973 

10 J5 

1098 

1161 

1225 

1289 

1354 

1419 

1481 

1550 

J616 

1684 

1751 

1 

974 

1086 

1099 

1168 

1226 

1290 

1355 

142o 

1485 

1551 

1618 

16^5 

1752 

2 

975 

1037 

1100 

1164 

1227 

1291 

1356 

142 

1486 

1552 

1619 

1 686 

1753 

8 

976 

1038 

1101 

1165 

1228 

1292 

1357 

1422 

1487 

1553 

1620 

1G87 

1755* 

4 

977 

1039 

1102 

1166 

1229 

1293 

1358 

1428 

1488 

1554 

16/1 

H 88 

1756 

5 

978 

1041 

1103 

1.67 

123o 

1 ?95 

1359 

1424 

1490 

1556 

1622 

1689 

1757 

6 

979 

104/ 

1105 

1168 

1262 

1296 

136" 

1425 

1491 

1557 

1623 

1690 

1758 

7 

980 

1043 

1106 

1169 

1238 

1297 

1361 

1426 

1492 

1558 

1624 

1692 

1759 

8 

981 

1044 

1107 

1 170 

1234 

1298 

1362 

1427 

1493 

1559 

1625 

1693 

1760 

9 

982 

lo45 

1108 

1171 

1286 

1299 

1363 

14/8 

1494 

1560 

16/6 

1694 

1761 

10 

988 

1046 

1109 

1172 

1236 

1300 

1364 

1430 

14 05 

1561 

1628 

1695 

1762 

11 

984 

1»47 

1110 

1173 

1237 

1301 

1366 

1431 

1496 

1562 

1629 

1696 

1764 

12 

985 

1048 

1111 

1174 

1238 

1302 

1367 

1482 

1497 

1 ‘ 63 

16:'0 

1697 

1765 

13 

986 

1049 

1112 

1175 

1239 

1303 

1368 

1438 

1498 

1564 

1631 

1698 

176G 

14 

987 

1050 

1113 

1176 

1240 

1304 

1869 

1434 

1499 

1565 

1632 

1699 

1767 

15 

98S 

1051 

1114 

1177 

1241 

1305 

1370 

1435 

1500 

1567 

1633 

170m 

1768 

16 

989 

1052 

1115 

1178 

1242 

1306 

1871 

1436 

15o2 

156' 

1634 

1701 

1769 

17 

990 

1053 

1116 

1179 

1248 

1807 

1372 

1437 

1503 

1569 

1635 

1703 

177o 

18 

991 

1054 

1117 

1181 

1244 

1308 

1373 

1438 

1504 

1570 

1637 

1704 

1772 

19 

893 

1055 

1118 

1182 

1245 

1310 

1374 

1439 

1505 

1571 

1638 

1705 

1773 

20 

994 

1056 

1119 

1183 

1246 

1311 

1375 

1440 

1506 

1572 

1639 

1706 

1774 

21 

995 

1057 

1120 

1184 

1248 

1312 

1376 

1441 

1507 

1573 

1640 

17o7 

1775 

22 

996 

1058 

1121 

1185 

1249 

1313 

1377 

1443 

1508 

1574 

1641 

1708 

1776 

23 

997 

1059 

1122 

1186 

1250 

1314 

1379 

1444 

1509 

1578 

1642 

17oy 

1777 

24 

998 

1060 

1123 

1187 

1251 

1815 

1380 

1445 

1510 

1577 

1643 

1711 

1778 

25 

999 

1061 

1125 

1188 

1252 

1316 

1381 

1446 

1511 

1578 

1644 

1712 

1780 

26 

iooo 

1**6 < 

1126 

1189 

1253 

1317 

1382 

1447 

1513 

1579 

1645 

1713 

1781 

27 

iooi 

1064 

1127 

1190 

1254 

1318 

1383 

1448 

1514 

1580 

1647 

17.4 

1782 

28 

1002 

1065 

1128 

1191 

1255 

1319 

1384 

1449 

1515 

1581 

1648 

1715 

1783 

29 

1003 

1066 

1129 

1192 

1256 

1320 

1385 

1450 

1516 

1582 

1649 

1716 

1784 

30 

1004 

1067 

1130 

1193 

1257 

1321 

1386 

1451 

1517 

1583 

1650 

1717 

1785 

31 

1005 

1068 

1131 

1194 

1258 

1322 

1387 

1452 

1518 

1584 

1651 

1718 

1786 

32 

1006 

1069 

1132 

1195 

1259 

1324 

13-8 

1483 

1519 

1585 

1652 

1720 

1787 

; 33 

1007 

107o 

1133 

1196 

1260 

1325 

1389 

1455 

1520 

1586 

1653 

1721 

1789 

34 

1008 

1071 

1134 

1198 

1261 

1326 

1890 

1456 

1521 

1588 

1654 

1722 

1790 

35 

1009 

1072 

1135 

1199 

1262 

1327 

1392 

1457 

1522 

1589 

1656 

1723 

1791 

36 

1010 

1073 

1136 

1200 

1264 

1328 

1393 

1458 

1524 

1590 

1657 

1724 

1792 

37 

1011 

1074 

1137 

1201 

1265 

1829 

1394 

1459 

1525 

1591 

1658 

1725 

1793 

3< 

1012 

1075 

1138 

1202 

1266 

1330 

1395 

1460 

1526 

1592 

1659 

1726 

1794 

I 39 

1013 

1076 

1139 

1208 

1267 

1331 

1396 

1461 

1527 

1593 

1660 

1727 

1795 

40 

1014 

3 077 

1140 

1204 

1268 

1 332 

1397 

1462 

1528 

1594 

1661 

1729 

1797 

41 

1015 

lo78 

1141 

1205 

1269 

1333 

1398 

1463 

1529 

1595 

1662 

1780 

1798 

42 

1016 

1079 

1142 

1206 

127o 

1334 

1399 

1464 

153o 

1596 

1668 

1731 

1799 

43 

1018 

1080 

1144 

1207 

1271 

1335 

14' 10 

1465 

1531 

1598 

1664 

1732 

1800 

44 

1019 

1081 

1145 

1208 

1272 

1336 

1401 

1467 

1532 

1599 

1666 

1733 

1801 

45 

1020 

1082 

1146 

1209 

1278 

1338 

1402 

1468 

1533 

1600 

1667 

1734 

1802 

46 

1021 

1084 

1147 

121o 

1274 

1339 

1403 

1469 

1538 

1601 

1668 

1735 

1803 

47 

1022 

1085 

1148 

1211 

127;' 

1340 

1405 

147o 

1536 

1602 

1669 

1736 

1805 

48 

1023 

1"86 

1149 

1212 

1276 

1841 

1406 

1471 

1537 

16 03 

167o 

1737 

1806 

49 

1024 

1087 

1150 

1213 

1277 

1342 

14o7 

1472 

1538 

1604 

1671 

1739 

1807 

50 

1025 

1088 

1151 

1215 

1278 

1343 

1408 

1473 

1539 

1605 

1672 

1740 

1808 

51 

1026 

1089 

1152 

1216 

128* 

1344 

1409 

1471 

1540 

1606 

1678 

1741 

1809 

52 

1027 

1090 

1158 

1217 

1281 

1845 

141m 

1475 

1541 

1608 

1675 

1742 

1810 

53 

1028 

1091 

1154 

1218 

1282 

1346 

1411 

1476 

1842 

1609 

1676 

1743 

1811 

54 

1029 

1092 

1155 

1219 

1288 

1847 

1412 

1477 

1518 

1610 

1677 

1741 

1813 

55 

1030 

1093 

1156 

122o 

1284 

1348 

1413 

1479 

1544 

1611 

167^ 

174 6 

1814 

5 6 

103: 

1094 

1157 

1221 

1285 

1349 

1414 

14 80 

1546 

1612 

1679 

1747 

1815 

57 

1032 

1095 

1158 

122 

1286 

1350 

1415 

1481 

1547 

1613 

168' 

1748 

1816 

58 

1038 

1096 

1159 

1220 

1287 

1852 

1416 

1482 

1548 

1611 

1681 

1749 

1817 

59 

1<:>34 

1097 

1 I60 

12/4 


1358 

14i 8 

1483 

1548 

1615 

16'2 

1750 

1818 




























































——B— — H —— « 

Meridianal Parts. TABLE IV. 1 


9 

rr 

80° 

31° 

32° 

33° 

34° 

35° 

CO 

37° 

3S° 

39° 

° 0 

41° 

0 

1819 

1*88 

1958 

2028 

210 

2171 

2214 

2318 

2393 

2468 

2545 

2623 

2702 

1 

1821 

1890 

1959 

2030 

2101 

2173 

2246 

2319 

2394 

2470 

2546 

2624 

2703 1 

2 

1822 

1891 

I960 

203; 

2102 

2174 

2247 

2320 

2395 

2471 

2548 

2625 

2704 j 

3 

1823 

1892 

1962 

2082 

2103 

2175 

2248 

2322 

2396 

2472 

2549 

2627 

2706 ! 

4 

1824 

1893 

1968 

2033 

2101 

2176 

2249 

2323 

239S 

2473 

2550 

2628 

2707 

5 

1825 

1894 

lit 61 

2034 

2105 

2178 

2250 

2324 

2399 

2475 

2551 

2629 

2708 

6 

1826 

1895 

1965 

2(‘35 

2107 

2179 

2252 

2325 

2400 

2476 

2553 

2631 

2710 

7 

1827 

1896 

1936 

2037 

2108 

218 • 

2253 

2327 

2401 

2477 

2551 

2632 

2711 

8 

1829 

1898 

1 .'67 

2038 

2109 

2181 

2254 

2328 

2403 

2478 

2555 

2633 

2712 

9 

1830 

1899 

1969 

2"39 

21K> 

2182 

2255 

2329 

2404 

248 

2557 

2634 

2714 

10 

1831 

1900 

197o 

2“40 

2111 

2181 

2257 

2330 

2405 

2481 

2558 

2636 

2715 

11 

1832 

1901 

1971 

2"41 

2113 

2185 

2258 

2332 

2406 

2482 

2559 

2687 

2716 

12 

1833 

1902 

1972 

2043 

21 4 

2186 

2259 

2333 

2468 

2484 

2560 

2638 

2718 

13 

1834 

1903 

1973 

2044 

2115 

2187 

2260 

2334 

2409 

2485 

2562 

264" 

2719 

14 

1835 

19(65 

1974 

•2045 

2116 

2188 

2261 

2335 

2410 

2486 

2563 

2641 

2720 

15 

1837 

1906 

1976 

2046 

2117 

2190 

2263 

2337 

2411 

24*7 

2564 

2642 

2722 

10 

1838 

1907 

1977 

2047 

2119 

2191 

2264 

2338 

2413 

2489 

256G 

2644 

2723 

17 

1839 

1908 

1978 

2< '4 s 

212 

2192 

2265 

2339 

2414 

2490 

2567 

2645 

2724 

18 

1840 

1909 

1979 

2050 

2121 

2193 

2266 

2340 

2415 

2491 

2568 

2946 

27 2G 

10 

1841 

191o 

19So 

2051 

2122 

2194 

2268 

2342 

2416 

2492 

2569 

264S 

2727 

20 

1842 

1912 

1981 

2052 

2123 

2196 

2269 

2343 

2418 

2494 

2*71 

2«49 

2728 

21 

1343 

1913 

1983 

2053 

2125 

2197 

227" 

2344 

2419 

2495 

2572 

285" 

2729 

22 

1S45 

1914 

1984 

2054 

2126 

2198 

2271 

2345 

2420 

2496 

2578 

2655 

2731 

23 

1846 

1915 

1985 

2o56 

2127 

2’. 99 

2272 

2346 

2422 

2498 

2 75 

2653 

2732 

24 

1S47 

1916 

1936 

2057 

2128 

22(>0 

2274 

2348 

2423 

2499 

2576 

2654 

2738 

25 

1848 

1917 

1987 

2058 

2121' 

2202 

2275 

2349 

2424 

2300 

2577 

2655 

2735 

20 

1849 

191* 

198> 

2059 

2131 

2203 

2276 

2350 

2425 

2501 

2578 

2657 

2736 

27 

1850 

1920 

199o 

2060 

2132 

2204 

2277 

2351 

2427 

2508 

2580 

2658 

2737 

28 

1852 

1921 

1991 

2061 

2133 

2205 

2279 

2353 

2428 

2504 

2581 

2659 

2739 

29 

1853 

1922 

1992 

2063 

2134 

2207 

228" 

2354 

2429 

2505 

2582 

2661 

‘4740 

30 

1854 

1923 

1993 

2064 

2135 

2208 

2281 

2355 

2430 

2506 

2584 

2662 

2742 

31 

1855 

1924 

1994 

206 

2137 

2209 

2282 

2356 

2432 

2508 

25*5 

2663 

2743 

32 

1856 

1925 

1995 

2066 

2138 

2210 

2283 

2358 

2433 

2509 

2586 

2665 

2744 

a 8 

1857 

1927 

1997 

2<‘67 

2139 

2211 

2285 

2859 

2434 

2510 

2588 

2666 

2746 

34 

1858 

1928 

1998 

2069 

214" 

221 

228" 

2860 

213 

2512 

2589 

2667 

2747 

85 

I860 

1929 

1999 

2«'»7 o 

2141 

2214 

22>7 

2361 

2437 

2513 

2590 

2669 

2743 i 

30 

1<61 

1930 

2"0<* 

2071 

2143 

2215 

2288 

2368 

2438 

2514 

2591 

2670 

2750 

37 

186' 

1931 

2001 

2072 

2144 

22 6 

2290 

2864 

2439 

2515 

2593 

2671 

2751 

38 

1863 

19'>2 

2o02 

2o78 

2145 

2217 

2291 

2365 

244" 

2517 

2594 

267 

2752 1 

39 

1864 

1934 

2004 

2u7 o 

2146 

2219 

2292 

2366 

2442 

2518 

2595 

2674 

2754 

40 

1865 

1935 

2005 

2o7o 

2.147 

9220 

2293 

2 r 'G8 

2443 

2519 

2597 

2675 

2755 

41 

1866 

1936 

2006 

2077 

2149 

2221 

2 95 

2369 

2444 

2521 

2598 

2676 

2756 

42 

1868 

1937 

2007 

2o78 

2150 

2222 

2296 

2370 

2445 

2522 

2599 

2678 

2758 

43 

1869 

1938 

2008 

2079 

2151 

2224 

2297 

2371 

2447 

2523 

2601 

2679 

2759 

44 

1870 

1939 

2010 

2080 

2152 

; 225 

2298 

2373 

2448 

2524 

26o2 

268o 

2760 

45 

1871 

1941 

2oll 

2082 

2153 

2226 

2299 

2374 

2449 

2526 

26o3 

2682 

2762 

46 

1872 

1942 

2012 

2083 

2155 

2227 

2301 

2875 

2451 

2527 

2604 

2688 

2763 

47 

1873 

1943 

2013 

2084 

2156 

2228 

2302 

2376 

2452 

252? 

2606 

2681 

2764 

48 

1875 

1944 

2014 

2085 

2157 

223" 

2303 

2378 

2453 

2530 

2607 

2686 

2766 

49 

1876 

1945 

2015 

2086 

2158 

2231 

2304 

2379 

2454 

2531 

2608 

2687 

2767 

50 

1877 

1946 

2017 

208 v 

2159 

2232 

2306 

2380 

2456 

2582 

2610 

2688 

2768 

51 

1878 

1948 

2018 

2089 

2161 

2233 

2807 

2381 

2457 

2533 

2611 

2690 

2770 

52 

1879 

1949 

2019 

2090 

2162 

2235 

2808 

2383 

2458 

2535 

2612 

2691 

2771 

53 

1880 

1950 

2020 

2091 

2163 

2236 

2309 

2381 

2459 

2536 

2614 

2692 

2772 

54 

1881 

1951 

2021 

2092 

2164 

2237 

2311 

2885 

2461 

2537 

2615 

2694 

2774 

55 

1883 

1952 

2022 

20 4 

2165 

2238 

2312 

2886 

2462 

253S 

26; 6 

2695 

2775 

56 

1-84 

1958 

2024 

209.* 

2167 

2239 

2313 

238 s1 

2463 

2540 

2617 

2696 

2776 

57 

1885 

1955 

2025 

2096 

2168 

224 

2314 

2389 

2464 

2541 

2619 

2698 

2778 

58 

1886 

1956 

2026 

2097 

2169 

2212 

2316 

2890 

2466 

2542 

262" 

2699 

2779 

59 

18n7 

1957 

2027 

2('«8 

2170 

224: 

2317 

2391 

2467 

2544 

2G21 

2700 

2700 










































TABLE IV. 


Mer'di.i! al Farts. 


119 


• 

4*?o 

43° 

44° 

45° 

46° 

47° 

4S° 

49° 

50° 

61° 

52° 

53° 

54° 

0 

2782 

2863 

2940 

3030 

3116 

3208 

8292 

3382 

8474 

3569 

366. 

3764 

365 

1 

2783 

2864 

2947 

3031 

3117 

8v04 

3293 

3384 

847 6 

3570 

3661 

3, 65 

3S66 

2 

2784 

2860 

2949 

3033 

3118 

3-06 

329. 

3385 

3478 

3572 

3668 

3. 67 

3; 68 

3 

2736 

2867 

2950 

3034 

3120 

3207 

3296 

3387 

3479 

3573 

3670 

3. 69 

3870 

4 

2787 

2809 

2951 

3036 

3121 

8209 

3298 

3388 

3481 

3575 

3672 

3.70 

3871 

5 

2788 

2870 

2953 

3017 

3123 

3210 

3299 

3390 

8482 

3577 

oG* o 

3.72 

3873 

6 

2790 

2871 

2954 

3038 

3125 

3212 

3801 

3391 

8484 

3578 

3678 

3.74 

3375 

7 

2791 

2873 

2956 

3040 

8126 

3213 

330i 

8893 

3485 

3580 

3677 

3775 

3877 

8 

2792 

2874 

2957 

3041 

3127 

8214 


3394 

3487 

85S2 

3678 

3.77 

3878 

9 

2794 

2875 

2958 

3043 

3129 

3216 

3305 

3396 

8488 

3583 

3680 

3779 

3880 

' 10 

2795 

2877 

2960 

3044 

3130 

3217 

3306 

3397 

3490 

3585 

3681 

3780 

3382 

11 

2797 

2878 

2961 

3046 

8131 

3219 

3308 

3399 

8492 

3586 

3683 

3782 

3883 

, 12 

2798 

28SO 

2963 

3047 

3183 

3220 

3309 

3400 

3493 

3588 

8685 

3/84 

3885 

13 

2799 

2881 

2964 

3048 

3134 

3222 

3311 

3402 

8495 

8590 

3686 

3785 

8887 

14 

2801 

2882 

2965 

3050 

8136 

3224 

8312 

3403 

3496 

8591 

368i 

3787 

3889 

15 

2802 

2884 

2967 

3051 

3137 

3225 

3314 

3405 

3498 

3593 

3690 

3789 

3890 

10 

2803 

2885 

2968 

3058 

3189 

3226 

3316 

3407 

3499 

8594 

3691 

3790 

3892 

17 

2805 

2886 

2970 

3054 

3140 

3228 

3817 

3408 

3501 

3596 

8693 

3792 

3894 

18 

2806 

2888 

2971 

3055 

3142 

3229 

3319 

8410 

3503 

3598 

3695 

3794 

3895 

19 

2807 

28S9 

2972 

8057 

3143 

3231 

3820 

3411 

3504 

3599 

3696 

3795 

3897 

20 

2809 

2891 

2974 

3058 

3144 

3232 

3322 

3413 

3506 

3601 

3698 

3797 

3899 

21 

2810 

2892 

2975 

3060 

3146 

3234 

00 60 

8414 

3507 

3602 

3699 

3799 

3901 

22 

2811 

2893 

2976 

3061 

3147 

3235 

3325 

3416 

3509 

3604 

37ul 

3800 

3902 

23 

2813 

2895 

2978 

3063 

8149 

3237 

3326 

3417 3510 

3606 

8 i 08 

3802 

3904 

24 

2814 

2896 

2979 

3064 

3150 

3238 

3328 

3419 

3512 

8607 

3704 

3S04 

3906 

25 

2815 

2897 

2981 

3065 

3152 

3240 

8829 

8420 

3514 

8609 

3706 

38' 6 

3907 

20 

2817 

2S99 

2982 

3067 

3153 

3241 

8331 

8422 

3515 

3610 

3708 

3-07 

3909 

27 

2813 

2900 

2983 

3068 

3155 

3242 

3332 

3423 

3517 

3612 

3709 

3809 

3911 

28 

2820 

2902 

2985 

8070 

3156 

3244 

3384 

3425 

3518 

3614 

3711 

3811 

3913 

29 

2821 

2903 

2986 

3071 

3157 

3245 

3335 

3427 

3520 

3615 

3718 

3812 

3914 

30 

9822 

2904 

2988 

3073 

3159 

3247 

3337 

3428 

3521 

3617 

3714 

3814 

3916 I 

31 

2824 

2906 

2989 

3074 

3160 

3248 

3338 

3430 

3523 

3618 

3716 

3816 

3918 

32 

2825 

2907 

2991 

3075 

3162 

3250 

3340 

3431 

3525 

8620 

3717 

3817 

3919 

33 

2826 

2908 

2992 

3077 

3163 

3251 

3341 

3433 

3526 

3622 

3719 

3819 

3921 

34 

2828 

2910 

2-i93 

3078 

3165 

8253 

3343 

3434 

3528 

3i)2o 

3721 

3321 

3923 

35 

2829 

2911 

2995 

3080 

3166 

3254 

3344 

3436 

3529 

3625 

3l ^'6 

3822 

3925 

! 36 

2830 

2913 

2996 

3081 

3168 

3256 

3346 

3437 

3531 

3626 

3.2 4 

3824 

3926 

37 

2832 

2914 

2998 

3083 

3169 

3257 

3347 

3439 

3532 

3.528 

3726 

3826 

3928 

38 

2833 

2915 

2999 

3084 

3171 

3259 

3349 

3440 

3534 

3o£J0 

3727 

3827 

3.30 

39 

2834 

2917 

3000 

3085 

3172 

3260 

3350 

3442 

3536 

3631 

3729 

3829 

3932 

40 

2836 

2918 

3002 

30S7 

3173 

3262 

3352 

3443 

3537 

3683 

8731 

3831 

3933 

41 

2837 

2919 

3< >03 

30S8 

3175 

3263 

3353 

3445 

3539 

3634 

3732 

3832 

3935 

42 

2839 

2921 

3005 

3090 

3176 

3265 

3355 

3447 

3540 

3636 

3734 

3834 

3937 

43 

2840 

2922 

3006 

3091 

3178 

3266 

3356 

344S 

3542 

3638 

3736 

3836 

3938 

44 

2841 

2924 

3007 

3093 

8179 

3268 

3358 

3450 

3543 

3639 

3737 

3838 

3940 

45 

2843 

2925 

3009 

3094 

3181 

3269 

3359 

3451 

3545 

3(541 

3739 

3839 

3942 

40 

.2844 

29n6 

3010 

3095 

3182 

3271 

3361 

3453 

3547 

3643 

3741 

3841 

3244 

47 

2845 

29v8 

301‘4 

8097 

3184 

3272 

3362 

3454 

3548 

3644 

3742 

3843 

3945 

48 

2847 

2929 

8013 

3098 

3185 

8274 

3364 

3456 

3550 

8646 

3744 

3844 

3947 

49 

2348 

2981 

3014 

8100 

3187 

3275 

3365 

3457 

3551 

3647 

3746 

3846 

3949 

50 

2849 

2932 

3016 

3101 

3188 

3277 

3367 

3459 

3553 

3649 

3747 

3848 

3951 

51 

2851 

2933 

3017 

3103 

3190 

3278 

3368 

3460 

3555 

3651 

3749 

3849 

3952 

52 

2852 

2935 

3019 

3104 

3191 

3280 

3370 

3462 

3556 

3652 

3750 

3851 

3954 

53 

2854 

2936 

3020 

3105 

3192 

3281 

3371 

3464 

3558 

3654 

3752 

3853 

3956 

54 

2855 

2937 

3021 

3107 

3194 

3283 

3373 

3465 

3559 

3655 

3754 

3854 

3958 

5 

2856 

2939 

3023 

3108 

3195 

3284 

3374 

3467 

3561 

3657 

3755 

3856 

3959 

56 

2858 

2940 

3024 

3110 

3197 

8286 

3376 

3468 

3562 

3659 

3757 

3858 

3961 

57 

2859 

2942 

3026 

3111 

3198 

3287 

3378 

3470 

3564 

3660 

3759 

3360 

3963 

58 

2860 

2943 

3027 

3113 

3200 

3289 

3379 

8471 

3566 

3662 

3760 

3861 

3964 

59 

2862 

2944 

3029 

3114 

3201 

3290 

3381 

3473 

3567 

3664 

3762 

3863 

3966 

















































tk»a 


120 Merdianal Tarta. TABLE IV. 


i 

1 ' 

1 

55° 

56° 

57° 

58° 

59° 

60° 

61* 

62 tt 

63° 

64° 

65° 

66° 

1 

-J 1 

O 1 

i 0 

3968 

4074 

4183 

4294 

4409 

4527 

4649 

4775 

4905 

5039 

5179 

5324 

5474 | 

1 

3970 

4076 

4184 

4296 

4411 

4529 

4651 

4777 

4907 

5042 

5181 

5326 

5477 

2 

3971 

4077 

4186 

4298 

4413 

4531 

4653 

4779 

4909 

5044 

5184 

5328 

5479 

3 

3973 

4079 

4188 

4300 

4415 

4533 

4655 

4781 

4912 

5046 

5186 

5331 

5482 

4 

3915 

4081 

4190 

4302 

4417 

4535 

4657 

4784 

4914 

6049 

5188 

6333 

5484 

5 

3977 

4083 

4192 

4304 

4419 

4537 

4660 

4786 

4916 

6051 

5191 

5336 

5487 

6 

3978 

4085 

4194 

4306 

4421 

4539 

4662 

4788 

4918 

5053 

5193 

6338 

5489 

7 

3980 

4086 

4195 

4308 

4423 

4541 

4664 

4790 

4920 

5055 

5195 

5341 

5492 

8 

3982 

4088 

4197 

4309 

4425 

4543 

4666 

4792 

4923 

5058 

5198 

5343 

5495 

9 

3984 

4090 

4199 

4311 

4427 4545 

4668 

4794 

4925 

6060 

5200 

5346 

5497 

1C 

| 

3985 

4092 

4202 

4313 

4429 

4547 

4670 

4796 

4927 

5062 

5203 

5348 

5500 

11 

3987 

4094 

4203 

4315 

4431 

4549 

4672 

4798 

4929 

5065 

5205 

5351 

5502 

12 

3989 

4095 

4205 

4317 

4433 

4551 

4674 

4801 

4931 

5067 

5207 

5353 

5505 

13 

3991 

4097 

4207 

4319 

4434 

4553 

4676 

4803 

4934 

5069 

5210 

5356 

5507 

14 

3992 

4099 

4208 

4321 

4436 

4555 

4678 

4805 

4936 

5071 

5212 

5358 

5510 

15 

3994 

4101 

4210 

4323 

4438 

4557 

4680 

4807 

4938 

5074 

5214 

5361 

5513 1 

16 

3996 

4103 

4212 

4325 

4440 

4559 

4682 

4809 

4940 

5076 

5217 

5363 

5515 

17 

3998 

4104 

4214 

4327 

4442 

4562 

4684 

4811 

4943 

5078 

5219 

5366 

5518 ! 

18 

3999 

4106 

4216 

4328 

4444 

4564 

4687 

4814 

4945 

5081 

5222 

5368 

5520 j 

19 

4001 

4108 

4218 

4330 

4446 

4566 

4689 

4816 

4947 

5083 

5224 

5371 

5523 I 

20 

4003 

4110 

4220 

4332 

4448 

4568 

4691 

4818 

4949 

5085 

5226 

5373 

5526 

21 

4005 

4112 

4221 

4334 

4450 

45/0 

4693 

4820 

4951 

5088 

5229 

5376 

5528 

22 

4006 

4113 

4223 

4336 

4452 

4572 

4695 

4822 

4954 

5090 

5231 

5378 

5531 

23 

4008 

4115 

4225 

4338 

4454 

4574 

4697 

4824 

4956 

5092 

5234 

5380 

5533 

24 

4010 

4117 

4227 

4340 

4456 

4576 

4699 

4826 

4958 

5095 

5236 

5383 

5536 

ZD 

4012 

4119 

4229 

4342 

4458 

4578 

4701 

4829 

4960 

5097 

6238 

5385 

5539 

26 

4014 

4121 

4231 

4344 

4460 

4580 

4703 

4831 

4963 

5099 

5241 

5388 

5541 

27 

4015 

4122 

4232 

4346 

44C2 

4582 

4705 

4833 

4965 

5102 

5243 

53,0 

5544 I 

[ 28 

4017 

4124 

4234 

4347 

4464 

4584 

4707 

4835 

4967 

5104 

5246 

5393 

5546 

29 

4019 

4126 

4236 

4349 

4466 

4586 

4710 

4837 

4969 

5106 

5248 

5395 

5549 ! 

30 

4021 

4128 

4238 

4351 

4468 

4588 

4712 

4839 

4972 

5108 

5250 

5398 

5552 : 

! 31 

4022 

4130 

4240 

4353 

4470 

4590 

4714 

4842 

4974 

5111 

5253 

5401 

5554 j 

32 

4024 

4132 

4242 

4355 

4472 

4592 

4716 

4814 

4976 

5113 

525o 

5403 

5557 

33 

4026 

4133 

4244 

4357 

4474 

4594 

4718 

4846 

4978 

5115 

5258 

5406 

5559 1 

34 

4028 

4135 

4246 

4359 

4476 

4596 

4720 

4848 

4981 

5118 

5260 

5408 

5562 

i 35 

4029 

4137 

4247 

4361 

4478 

4598 

4722 

4850 

4983 

5120 

5263 

5411 

5565 ; 

36 

4031 

4139 

4249 

4363 

4480 

4600 

4724 

4852 

4985 

5122 

5265 

5413 

5567 

37 

4033 

4141 

4251 

4365 

4482 

4602 

4726 

4855 

4987 

5125 

5267 

5416 

5570 | 

38 

4035 

4142 

4253 

4367 

4484 

4604 

4728 

4857 

4990 

5127 

5270 

5418 

5573 ! 

39 

4037 

4144 

4255 

4369 

4486 

4606 

4731 

4859 

4992 

5129 

5272 

5421 

5575 i 

AO 

4038 

4146 

4257 

4370 

4488 

4608 

4733 

4861 

4994 

6132 

5275 

5423 

5578 

41 

4010 

4148 

4259 

4372 

4490 

461" 

4735 

4863 

4996 

5134 

5277 

5426 

5580 

42 

4042 

4150 

4260 

4374 

4492 

4612 

4736 

4865 

4999 

5136 

52'0 

5428 

5583 

43 

4044 

4152 

4262 

1376 

4494 

4614 

4739 

486^ 

5001 

5139 

5282 

5431 

5586 

44 

4045 

4153 

4264 

4378 

4495 

4616 

4741 

4*7" 

5003 

5141 

5284 

5433 

5588 

45 

4047 

4155 

4266 

4380 

4497 

4618 

4743 

4872 

5005 

5143 

52s7 

5436 

5591 

46 

4019 

4157 

4268 

4382 

4499 

4620 

4745 

4874 

5008 

5146 

5289 

5438 

5594 

47 

4051 

4159 

4270 

4384 

4501 

4623 

4747 

4876 

5010 

5148 

5292 

5441 

5596 

48 

4052 

41**1 

4272 

4386 

4503 

4625 

4750 

4S79 

5012 

5151 

5294 

5443 

5599 

49 

4054 

4162 

4274 

4388 

4505 

4627 

4752 

4881 

5014 

5153 

5297 

6446 

5602 

50 

4056 

4164 

4275 

4390 

4507 

4629 

4754 

4883 

5017 

5155 

5299 

5448 

5604 

51 

4058 

4166 

4277 

4392 

4509 

4631 

4756 

4885 

5019 

5158 

5301 

5451 

5607 

52 

4060 

4168 

4279 

4394 

4511 

4633 

4758 

4887 

5021 

5160 

5304 

5154 

5610 

53 

4061 

4170 

4291 

4396 

4513 

4635 

4760 

4890 

5023 

5162 

5306 

5456 

5612 

54 

4063 

4172 

4283 

439S 

4515 

4637 

4762 

4892 

5026 

5165 

53"9 

5458 

5615 

55 

4065 

4173 

4285 

4399 

4517 

4639 

4764 

4S94 

5028 

5167 

5311 

5461 

5617 

56 

4067 

4175 

4287 

4401 

4519 

4641 

4766 

4896 

5030 

5169 

5314 

5464 

5620 

57 

4069 

4177 

4289 

4103 

4521 

4643 

4769 

4898 

5033 

5172 

5316 

5466 

5623 

58 

4070 

4179 

4291 

4405 

4523 

4645 

4771 

4901 

5035 

5174 

5319 

5469 

5625 

59 

4072 

4191 

4292 

4407 

452 

4647 

4773 

4903 

5037 

5176 

5321 

5471 

5628 

1 












































































| TABLE IV. 



Meridianal Parts. 





121 

/ 

68° 

09° 

70° 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

73° 

79° 

80° 

0 

5631 

5795 

5966 

6146 

6335 

6584 

6746 

6970 

7210 

7407 

7745 

8046 

8375 

1 

5633 

5797 

5969 

6149 

6338 

0538 

6749 

6974 

7214 

747* 

7749 

8051 

8381 

2 

5636 

5800 

5972 

6152 

6341 

6541 

6753 

6978 

7218 

7476 

7754 

8 >56 

8387 

3 

5639 

580 

5975 

6155 

6345 

6545 

6757 

6982 

7222 

7481 

7759 

8<>61 

8393 

4 

5642 

5806 

5978 

6158 

6348 

6548 

6760 

6986 

7227 

7485 

7764 

8067 

8398 

5 

5614 

5809 

5981 

6161 

6351 

6552 

6764 

6990 

7231 

7490 

7769 

8072 

8404 

6 

5646 

5811 

5984 

6164 

6354 

6555 

6768 

6994 

7235 

7494 

7774 

8077 

8410 

7 

5650 

5814 

5985 

6167 

6358 

6558 

6771 

6997 

7239 

7498 

7778 

8o83 

8416 

8 

5652 

5817 

5989 

6170 

6361 

6562 

0775 

7001 

7243 

7503 

7783 

8088 

8422 

9 

5655 

5820 

5992 

6173 

6364 

6565 

6779 

7005 

7247 

7507 

7788 

8093 

8427 

10 

5658 

582; 

5995 

6177 

6367 

6569 

6782 

7009 

7252 

7512 

7793 

8099 

8433 

11 

5660 

5-25 

5998 

618" 

6871 

6572 

6786 

7013 

7256 

7516 

7798 

8104 

8439 

12 

5663 

5828 

6001 

61-3 

6374 

6.576 

6790 

7017 

7260 

7£21 

7803 

8109 

8445 

13 

5666 

5831 

6004 

6186 

6377 

6579 

6793 

7021 

7264 

7525 

7808 

8115 

8451 

14 

5668 

5834 

6007 

6189 

6380 

6583 

6797 

7025 

7268 

7580 

7813 

812o 

8457 

15 

5671 

5837 

6010 

6192 

63-4 

6586 

6801 

7029 

7273 

7535 

7817 

8125 

8463 

16 

5674 

5839 

6013 

6195 

6387 

6590 

6804 

7033 

7277 

7539 

7821 

8131 

8469 

17 

5676 

5842 

6016 

6198 

6390 

6593 

6808 

7037 

7281 

7544 

7827 

8136 

8474 

18 

5679 

5845 

6. *19 

6201 

6394 

6597 

6812 

7041 

7285 

7548 

7832 

8141 

8480 

19 

6682 

5848 

6022 

6205 

6397 

0600 

6815 

7045 

7289 

7553 

7837 

8147 

8486 

2o 

5685 

5^51 

602' 

6208 

6400 

6603 

6819 

7048 

7294 

7557 

7842 

8152 

8492 

21 

6687 

5854 

6028 

6211 

6403 

6607 

6823 

7052 

7298 

7562 

7847 

8158 

8498 

22 

5690 

5856 

6031 

6214 

6407 

6010 

6826 

7056 

7802 

7566 

7852 

8163 

8504 

23 

5693 

5859 

6o34 

6217 

6410 

6614 

6830 

7060 

7306 

7571 

7857 

8168 

8510 

24 

5695 

5862 

6037 

6220 

6413 

6617 

6834 

7064 

7311 

7576 

7862 

8174 

8516 

25 

5698 

5865 

6040 

6223 

6417 

6621 

6838 

7068 

7315 

7580 

7867 

8179 

85*2 

26 

5701 

5868 

6043 

6226 

6420 

6624 

6841 

707* 

73P‘ 

7585 

7872 

8185 

8528 

27 

51 04 

5871 

6046 

6280 

6423 

6628 

6845 

7076 

7323 

7589 

7877 

8190 

8534 

28 

5706 

5874 

6049 

6233 

6427 

6631 

6849 

708o 

7328 

7594 

7882 

8196 

8540 

29 

5709 

5876 

6052 

6236 

643o 

6635 

6853 

7084 

7332 

7599 

7887 

8201 

8546 

30 

5712 

5879 

6055 

6239 

6433 

6639 

6856 

7088 

7336 

7 603 

7892 

8207 

8552 

31 

5715 

• r 882 

6058 

6242 

6437 

6612 

686o 

7092 

7341 

7 60s 

7897 

8212 

8558 

32 

5717 

5885 

6061 

6245 

6440 

6646 

6864 

7096 

7345 

7612 

7902 

8218 

8565 

33 

5720 

5888 

6o64 

6249 

6443 

6049 

6868 

7100 

7849 

7617 

79' 7 

8223 

8571 

34 

5723 

5891 

6067 

6252 

6447 

6653 

0871 

7101 

7353 

7622 

7912 

8229 

8577 

3-' 

5725 

5894 

6070 

6255 

6450 

6656 

6875 

7108 

7358 

7 626 

7917 

8234 

8583 

36 

5728 

5896 

6"73 

6258 

6453 

666o 

6879 

7112 

7362 

7631 

7922 

8240 

8589 

37 

5731 

589. 

6'»76 

6261 

6457 

6663 

6883 

7116 

736'' 

7636 

7s27 

8245 

8595 

3s 

5734 

59o2 

6079 

0264 

6460 

6667 

088" 

7120 

7371 

7640 

7932 

8251 

8601 

39 

5736 

5905 

6082 

6268 

6463 

6670 

6390 

7124 

7375 

7645 

7937 

8256 

8607 

40 

5739 

5908 

6085 

6271 

6467 

6674 

6394 

7128 

7379 

7650 

7942 

8262 

8614 

41 

5742 

5911 

6088 

6274 

647o 

6677 

6898 

7132 

7384 

7654 

7948 

8267 

8620 

42 

5745 

5914 

6091 

6277 

6473 

6681 

6901 

7136 

7388 

7659 

7953 

8273 

8626 

43 

5747 

5917 

6094 

6280 

6477 

6685 

6905 

7140 

7392 

7604 

7958 

8279 

8632 

44 

5750 

5919 

6097 

6283 

648o 

6688 

0909 

7145 

7397 

7668 

7963 

8284 

8638 

45 

5753 

5922 

6100 

6'87 

6183 

6692 

6913 

7149 

7401 

7673 

7968 

8290 

8644 

46 

5756 

6925 

6103 

6290 

6487 

6695 

6917 

7153 

7406 

7678 

7973 

8295 

8651 

47 

5758 

5928 

6106 

6293 

649o 

6699 

6920 

7157 

7410 

7683 

7978 

8301 

8657 

48 

5761 

5931 

6109 

6-296 

6494 

6702 

6924 

7161 

7414 

7687 

7983 

8307 

8663 

49 

5764 

5934 

6112 

6299 

6497 

6706 

6928 

7165 

7419 

7692 

7989 

8312 

8669 

50 

5767 

5937 

6115 

6308 

6500 

6710 

6932 

7169 

7423 

7697 

7994 

8818 

8676 

51 

5770 

5940 

6118 

6306 

6504 

6713 

6936 

7173 

7427 

7702 

7999 

8324 

8682 

52 

5772 

5943 

6121 

6309 

6507 

6717 

6940 

7177 

7432 

7706 

8004 

8329 

8688 

53 

5775 

594»! 

6124 

6312 

6511 

6720 

6943 

7181 

7436 

7711 

8009 

8335 

8695 

54 

5778 

5948 

6127 

6315 

6514 

6724 

6947 

7185 

7441 

7716 

8014 

8341 

8701 

55 

5781 

5951 

6130 

6319 

6517 

6728 

6951 

7189 

7445 

7721 

8020 

8347 

8707 

56 

5783 

5954 

6133 

6322 

6521 

6731 

6955 

7194 

7449 

7725 

8025 

8352 

8714 

57 

5786 

5957 

6136 

6325 

6524 

6735 

6959 

7198 

7454 

7730 

8030 

8358 

8720 

58 

5789 

5960 

6140 

6328 

6528 

6738 

6963 

7202 

7458 

7735 

8035 

8364 

8726 

59 

5792 

5963 

6143 

6382 

6531 

6742 

6966 

72o6 

7463 

7740| 

8o4n 

8369 

8733 












































Meridianal Parts. TABLE IV. 



9 

81* 

82° 

83* 

84* 

85° 



0 

8739 

9145 

9606 

10137 

10765 



1 

8745 

9153 

9614 

10146 

10776 

• 


2 

8752 

9160 

9622 

10156 

10788 


- 

8 

8758 

9167 

9631 

10166 

107‘-'9 



4 

8765 

9174 

9639 

10175 

10811 



6 

8771 

9-182 

9647 

10185 

10822 



6 

8778 

9189 

9655 

10195 

10S34 



7 

8784 

9197 

9664 

10205 

10846 



8 

8791 

9203 

9672 

10214 

10858 



9 

8797 

9211 

9683 

10224 

10869 



10 

8804 

9218 

9689 

10234 

10881 



n 

8810 

9225 

9697 

10244 

10893 



12 

8817 

9233 

9706 

10254 

10905 



13 

3823 

9240 

9714 

10264 

10917 

• „ ' 


14 

8830 

9248 

9723 

10273 

10929 



15 

8836 

9255 

9731 

10283 

10941 

. 


16 

8843 

9262 

974'i 

10293 

10953 



17 

8849 

9270 

9748 

103O3 

10965 

'. . «• 


18 

8856 

9277 

9757 

10314 

10978 

f -x ' \, " ’ 


19 

8863 

9285 

9765 

10324 

10990' 



20 

8869 

9292 

9774 

10834 

11002 

- * •: •• 


21 

8876 

9300 

9783 

10344 

11014 

• 


22 

8883 

9307 

9791 

10354 

11027 

:; y j 1 


23 

8889 

9315 

9800 

10364 

11039 

• • ' • $ : 1 


84 

8896 

9322 

9809 

10374 

11052 

1' . . »f 


25 

8903 

9330 

9817 

10385 

11064 

• . - 


26 

8909 

9337 

9826 

10395 

11077 

!;»* 


27 

8916 

9345 

9835 

10405 

11089 

; i'j t 


28 

8923 

9353 

9844 

10416 

11102 

-t 


29 

8930 

9360 

9852 

10426 

11115 

. ; u >. * 


30 

8936 

9368 

9861 

10437 

11127 

■ • j V- • 


31 

8943 

9376 

9870 

10447 

11140 

; ' • 


32 

8950 

9383 

9879 

10457 

11153 

, A .r 

* 

. 

33 

8957 

9391 

9888 

10468 

11160 

. * • i 


34 

8963 

9399 

9897 

10479 

11179 

v - ‘i. 


35 

8970 

9407 

9906 

10489 

11192 


- 

36 

8977 

9414 

9915 

10500 

11205 

'7*k 


37 

8964 

9422 

9924 

10510 

11213 



38 

8991 

9430 

9933 

10521 

11231 



S9 

8998 

9438 

9942 

10532 

11244 

i 


40 

9005 

9445 

9951 

10542 

11257 

. ; Ijji, 


41 

9012 

9453 

9960 

10553 

11270 

r ' \ '-**■’*.-*- 


42 

9018 

9461 

9969 

10564 

11284 

■ •** 


43 

9025 

9469 

9978 

10575 

11297 

A r , 


44 

9032 

9477 

9987 

10586 

11310 



45 

9039 

9485 

9996 

10597 

11324 



46 

9046 

9498 

10005 

10608 

11837 



47 

9053 

9501 

10015 

10619 

11351 



48 

9060 

9509 

10024 

10630 

11365 



49 

9067 

9517 

10033 

10641 

11378 

j * 


50 

9074 

9525 

10043 

10652 

11392 

r\ 


51 

9081 

9533 

10052 

10663 

11406 



52 

9088 

9541 

10061 

10674 

11420 



53 

9096 

9549 

10071 

10685 

11434 

\ ‘ . -t 


54 

9103 

9557 

10080 

10696 

11448 



55 

9110 

9565 

10089 

10708 

11462 



56 

9117 

9573 

10099 

10719 

11476 



57 

9124 

9581 

10108 

10730 

11490 



58 

9131 

9589 

10118 

10742 

11504 

* 


59 

9138 

9598 

10127 

10753 

11518 























TABLE V. 

Dip of the Sea Horizon. 


is, 
v: » 

d‘ 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 

29 

30 

31 
82 

33 

34 

35 


Din of ihe 
Horizon. 

- D - 

~ r-t- 

*5jO 

1 ' 

Mt-- 
O ^3 

3 o 

N —i 
*> r» 

B S - 
• O 

# /# 

V 

i t 

0 59 

38 

6 4 

1 24 

41 

6 18 

1 42 

44 

6 32 

1 58 

47 

6 45 

2 12 

50 

6 58 

2 25 

53 

7 10 

2 36 

56 

7 12 

2 47 

59 

7 24 

2 57 

62 

7 45 

3 07 

65 

7 56 

3 16 

63 

8 07 

3 25 

71 

8 IS 

8 33 

74 

8 28 

3 41 

77 

8 38 

3 49 

80 

8 48 

3 56 

83 

8 58 

4 04 

86 

9 08 

4 11 

89 

9 17 

4 17 

92 

9 26 

4 24 

95 

9 36 

4 8i 

98 

9 45 

4 37 

101 

9 54 

4 43 

104 

10 02 

4 49 

107 

10 11 

4 55 

110 

10 19 

5 01 

113 

10 23 

5 07 

116 

10 86 

5 13 

119 

10 44 

5 1* 

122 

10 52 

5 24 

125 

11 00 

5 2 

128 

11 08 

5 34 

131 

11 16 

5 39 

134 

11 24 

5 44 

137 

11 31 

5 49 

140 

11 39 


TABLE Vn. 

Mean Refraction of Celestial Objects. 


128 




TABLE VI. 

Dip of the Sea Hori¬ 
zon at different Dis¬ 
tances from it. 


Dist. 

in 

Miles. 

High t of Eye in Ft. 

l 

5 

10 

15 

20 

25 

30 


1 

/ 

11 

9 

22 

9 

34 

9 

45 

56 

# 

68 


4 

6 

11 

17 

22 

28 

34 

8 

i 

4 

8 

12 

15 

19 

23 


1 

4 

6 

9 

12 

15 

17 


n 

3 

5 

7 

9 

12 

14 


14 

3 

4 

6 

8 

9 

12 


2 

2 

3 

5 

6 

8 

10 


2* 

2 

3 

5 

6 

7 

8 

9 

3 

2 

3 

4 

5 

6 

7 


34 

2 

8 

4 

5 

6 

6 


4 

2 

3 

4 

4 

5 

6 


5 

2 

3 

4 

4 

5 

5 


6 

2 

O 

ol 

4 

4 

5| 

5 



Alt. 


0 

10 

20 

30 

40 


0 

It' 

K 

30 

40 

5< 


Refr. 


/ // 

33 0 
31 22 
29 50 
28 23 
27 00 
50 25 42 
24 29 
23 20 
22 15 
21 15 

20 18 
19 25 


10 

20 

80 

40 

50 
3 0 

10 


20 

80 

40 

50 


13 
13 
12 
12 
0 11 


10 

20 

80 

40 

50 

5 0 
10 
20 
80 
40 
50 
i 0 
lo 
20 
30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

80 

40 

50 

0 

10 

20 

30 

40 

50 


0 18 35 

17 4S 
17 04 
16 24 
15 45 
15 09 
14 34 
14 04 

34 
06 
40 
15 
51 
29 
0* 
48 
29 
11 


11 

11 

10 

10 

10 


9 54 
9 38 
9 23 
9 08 
8 54 
8 41 
8 28 
8 15 
8 03 
7 51 

40 

30 

20 

11 

02 

53 

4 

37 


6 29 
6 22 

6 15 
6 08 
6 01 
5 55 
5 48 
5 32 
5 36 


5 41 
5 25 
5 20 


19 


Alt. 

Refr. 

Alt. 

Refr 

Alt. 

Refr. 

Alt. 

Refr. 

» / 

/ // 

o / 

/ // 

O / 

/ // 

O 

// 

0 ( 

5 15 

20 C 

2 35 

22 ( 

l 81 

67 

24 

1C 

5 10 

It 

2 34 

4' 

1 29 

68 

23 

2< 

5 05 

2( 

2 32 

33 C 

1 2* 

69 

22 

3C 

5 00 

8( 

2 31 

2( 

1 26 

70 

21 

4( 

4 56 

40 

2 29 

4< 

1 25 

71 

19 

5C 

4 51 

5< 

2 2' 

34 0 

1 24 

72 

18 

1 ( 

4 47 

21 0 

2 27 

20 

1 23 

73 

17 | 

10 

4 43 

10 

2 26 

40 

1 22 

74 

16 

2< 

4 89 

20 

2 25 

35 0 

1 21 

75 

15 

30 

4 84 

30 

2 24 

20 

1 20 

76 

14 

4< 

4 31 

40 

2 23 

40 

1 19 

77 

13 

50 

4 27 

50 

2 2 

36 0 

1 18 

78 

12 

2 0 

4 23 

22 0 

2 20 

80 

1 17 

79 

11 ! 

10 

4 2< 

10 

2 19 

37 0 

1 16 

80 

10 

20 

4 16 

20 

2 18 

80 

1 14 

81 

9 

80 

4 13 

30 

2 17 

38 0 

1 13 

82 

8 1 

4n 

4 09 

40 

2 16 

80 

1 11 

83 

7 

50 

4 06 

50 

2 15 

39 0 

1 10 

84 

6 1 

3 0 

4 03 

23 0 

2 14 

30 

1 09 

85 

5 j 

10 

4 0o 

10 

2 13 

40 0 

1 08 

86 

4 

20 

3 57 

20 

2 12 

80 

1 07 

87 

8 

3<> 

3 54 

30 

2 11 

41 0 

1 05 

88 

2 

40 

3 51 

40 

2 10 

30 

1 04 

89 

1 

50 

3 48 

to 

2 09 

42 0 

1 03 

90 

0 

4 0 

3 45 

24 0 

2 08 

30 

1 02 



10 

3 43 

10 

2 07 

43 0 

1 01 



20 

3 40 

20 

2 06 

30 

1 00 



30 

3 38 

30 

2 05 

44 0 

0 59 



40 

3 85 

40 

2 0 4 

30 

0 58 



50 

3 33 

50 

2 03 

45 0 

0 57 



5 0 

3 30 

25 0 

2 02 

30 

0 56 



10 

3 28 

10 

2 01 

46 0 

0 55 



20 

3 26 

20 

2 00 

30 

0 54 



30 

3 24 

80 

1 59 

47 0 

0 53 



40 

3 21 

40 

1 58 

30 

0 52 



50 

3 19 

50 

1 57 

48 0 

0 51 



1 0 

3 17 

26 0 

1 56 

30 

0 50 



10 

3 15 

10 

1 55 

49 0 

0 49 



20 

3 12 

2o 

1 55 

30 

0 49 



30 

3 10 

30 

1 54 

50 0 

0 48 



40 

3 08 

40 

1 53 

30 

0 47 



50 

3 06 

50 

1 52 

51 0 

0 46 



0 

3 04 

27 0 

1 51 

80 

0 45 



10 

3 03 

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.. ‘ 

























ARE THE STANDARD, because they excel all others in 

THE FULLNESS OF THEIR 


ETYMOLOGIES, SYNONYMS MIND 

‘DEFINITIONS. ■ 


TN all these respects the SCHOOL DICTIONARIES, 
A as compared with other dictionaries of similar grade, are 
equally pre-eminent with the UNABRIDGED. A copy of 

Webster’s Primary Dictionary 
Webster’s Common-School Dictionary, or 

\ r 

Webster’s High-Scliool Dictionary 

Should be in the hands of every pupil in our public and private 

schools. 

We ask especial attention of teachers and High School and 
Academic pupils to the 

ACADEMIC DICTIONARY 

As in every way the best student’s dictionary aside from the 
Unabridged. In the matter of etymologies, synonyms and defini¬ 
tions it is beyond all comparison the best book for the pupil’s 
desk, discriminating clearly as to the origin, use, and office of 
every word. 

In view of all these facts, we confidently urge upon school 
officers the importance of making the use of Webster’s School 
Dictionaries in their schools as general as that of any other text¬ 
book on the list. 

Liberal terms will be made for the supply of these books 
for first introduction into schools, and for specimen copies of the 
various books for examination and the use of teachers. 

IVISON, BLAKEMAN & CO. 

Spencerian STEEL PENS 753 & 755 Broadway, New York 
• • are the best • » 149 Wabash Avenue, Chicago 












WEBSTER’S 

Unabridged Dictionary 

PUBLISHED BY 

G. & C. MERRIAM & CO., SPRINGFIELD, MASS. 

Webster’s School Dictionaries 

PUBLISHED BY 

IVISON, BLAKEMAN & CO., NEW YORK AND CHICAGO 

EMBRACING 

WEBSTER’S PRIMARY DICTIONARY 
WEBSTER’S COMMON-SCHOOL DICTIONARY 
WEBSTER’S HIGH-SCHOOL DICTIONARY 
WEBSTER’S ACADEMIC DICTIONARY 


WEBSTER’S DICTIONARIES have been officially recommended by the 
State Educational Authorities of nearly every State in the Union. 

THE LEADING SERIES of School Books published in this country, em¬ 
bracing over 25,000,000 Volumes published annually, are based upon Webster. 

From the earliest days of the Republic, Webster and Webster’s orthog¬ 
raphy have been the one practically universal element in American education. 

Thus Webster has been an important bond in uniting and keeping the American 
nation one people in their language, speech, and all the outward forms of thought. 

While in Great Britain, to quote an eminent authority, “ nearly every county 
has its local dialect, its peculiar words and forms,” our own vast and diversified 
country, with its local forms of self-government, its widely varied social types and 
industries, and, more than all, its ingathering of vast numbers from all nations and 
languages, easily assimilates all forms of speech, and maintains in universal and 
uniform purity the noble legacy of our language undefiled. 

Can any one doubt that this easy victory over the most varied and corrupting 
agencies is mainly due to Webster’s Dictionaries ? 

The attention of educators and of all friends of universal education is solicited to 
the importance of perpetuating this purity of American speech by maintaining the 
authority of Webster in every public and private school in the land. 

While it is of the highest importance that Webster’s Unabridged should be 
upon the teacher’s desk or reference table of every school-room, it is of equal impor¬ 
tance that Webster’s School Dictionaries should be placed in the hands of every 
child in the schools of the country. 

IVISON, BLAKEMAN & CO. 

SPENCERIAN STEEL pens 753 & 755 Broadway, New York 

• • are the best • • 149 Wabash Avenue, Chicago 









■A GOLD MEDA L was awarded to Professor SWINTON at the Paris Ex¬ 
position of 1878, as an author of School Text-i ooks, he being the 
only American A uthor thus highly honored. 


Standard Text-Books 

By 

Professor Swinton. 


SWINTON’S WORD-BOOK SERIFS. The only perfectly 
graded Series of Spel’ers ey r made, and the cheapest in the market. In use 
in more than 10,000 Schools. 

Word Primer. A Beginner’s Book in Oral and Written Spelling. 96 pages, 

Word-Book of Spelling. Oral and Written. Designed to attain prac¬ 
tical results in the acquisition of the ord nary English vocabulary, and t<\ 
serve as an introduc ion to Word Analysis. 154 pages. 

W r ord Analyst*. A Graded Class-book of English Derivative Words, with 
practical exercises in Spelling, Analyzing, Defining, Synonyms, and the 
Use of Words. 1 vol. 128 pages. 

SWINTON’S HISTORIES. These books have attained great pop¬ 
ularity. A new edition of the “Condensed” has just been issued, in which 
the work has been brought d->wn to the presenttime, and six colored maps have 
been added. 

Primary History of U. S. First Lessons in our Country’s History, 
bringing out the salient points, and aiming to combine simplicity with 
sense. 1 vol. square, fully illustrated. 

Condensed Seliool History of U. S. A Conden c ed School History 
of the United State q constructed for definite results in Recitation, and con¬ 
taining a new method of Topical Reviews. New edition, brought down to 
the present time. Illustrated with Maps, many of which are colored, 
Portraits and Illustrations. 1 vol. cloth. 300 pages. 

Outlines of tlie World’s History. Ancient, Mediaeval and Modern, 
with special reference to the History of Mankind. A most excellent work 
for the proper introduction of youth into the study of General History. 
1 vol., with numerous maps and illustrations. 500 pages, 12010. 

SWINTON’S GEOGU.lPHirtL COURSE. The famous “two 
book series,” the freshest, best graded, most beautiiul and cheapest Geographi¬ 
cal Course ever published. Of the lar e cities that have adopted Swinton’s 
Geographies, we mention Washington, D. C., Rochester, N. Y., Troy, N. Y 
Brooklyn, N. Y., New York City, Kingston, N.Y., Augusta, Me., Charles¬ 
ton, S. C , L \ncaster. Pa., Williamsport, Pa., M acon, Ga. 11 round numbers, 
they have been adopted in more than 1 .OOO Cities and Towns in all parts of 
the countrv, and have, with marked preference , been made the bas s of Pro¬ 
fessional Training in the Leading Normal Schools of the United States. 
Elementnry Course lu Geography. Designed as a class-book for 
primary and intermediate grades ; and as a complete Shorter Course for un¬ 
graded schools. 128 pages, 8vo. 

Complete Course in-Geography. Physical, Industrial and Political; 
with a special Geography for each State in the Union. Designed as a class- 
book for intermediate and grammar grades. 136 pages, 4to. 

The Maps in both books possess novel features of the highest practical 
value in education. 

SWINTON’S RAMBLES AMONG WORDS; Their Poetry, 
History and Wisdom. A Standard Work to all who love the riches of the 
Englisn Language. By William Swinton, M.A. Hands mely bound in flexible 
cloth and marbled edges. 302 pages. 

*** The above may be had, as a rule, from any bookseller ; but when not 
thus obtainable, we will supply them, transportation paid, at liberal rates. De¬ 
scriptive Circulars and Price Lists will be sent on applicatton. 

Very liberal terms for introduction, exchange and examination. 

IYISON, BLAKEMAN & COMPANY, Publishers, 

NEW YORK and CHICAGO. 






* £ As Familiar to the Schools of the United States as 

Household Words. 


Robinson’s Progressive Course 

OF 

MATHEMATICS. 

-- 

R OBINSON’S PROGRESSIVE COURSE OF 

MATHEMATICS, being the most complete and scientific course of 
Mathematical Text-books published, is more extensively used in t'.e Schools 
and Educational Institutions of the United States than any compe ing series. 

In its preparation two objects were kept constantly in view : First , to fur¬ 
nish a full and complete Series of Text-Books, which should be sufficientto give 
the pupil a thorough and practical business education ; Second, to secure that 
inielleitualculture without which the mere acquisition of book knowledge is 
almost worthless. 

All the improvements of the best modern Text-Books, as well as many new 
and original meth ds, and practical opera ions not found in other similar works, 
have been incorporated into these books, and no labor or expense has been 
spared to give to the public a clear, scientific, comprehensive and complete 
system, not incumbered with unnecessary theories, but combining and system¬ 
atizing real improvements of a practical and u. eful kind. 

--- 

% 

Robinson s Shorter Course. 

In order to meet a demand from many quarters for a Series of Arithmetics, 
few in number and comprehensive in character, we have published the above 
course, in TWO books in which Oral and Written Arithmetic is combined. 
These books have met with very great popularity, having been introduced into 
several of the largest cities in the United States. They are unusually hand¬ 
some in get-up, and are substantially bound in cloth. 


*+■' Descriptive Circulars and Price Lists will be forwarded to Teachers 
and Educationists on application. The most liberal terms will be made for in¬ 
troduction , exchange and examination. 

Ivison, Blakeman & Company, 

Publishers, 

New York and Chicago. 



















“Real Swan Quill ActionA 


The Celebrated Double Elastic 

Spencerian Steel Pens. 

• - 

"J'he ATTENTION of TEACHERS, SCHOOL 

Officers, and others, is called to the superior adaptability of the Spen¬ 
cerian Steel Pens to School use, possessing, as they do, the important peculiar- 
ties of 


Durability, 'Elasticity, and Evenness of Point. 


They are made by the best workmen in Europe, and of the best materials, 
and are selected with the utmost care. 

These Pens are used very largely in the Schools of the United States ; al¬ 
most entirely by the Commercial Colleges, and are especially commended by 
Professional Penmen and Pe 1 Artists. So popular have taey become 
that of the 

WELL-KNOWN No. i, more than 8,000,000 are annually sold. 

V V r • 

The Spencerian Pens are comprised in twenty numbers , differing in flex¬ 
ibility and fineness of point. 


We are also the Sole Agents for the United States for 

Perry & Co/s Steel Pens, 

which, in connection with our SPENCERIAN brand, gives us the control of 
the largest variety and best line of Ste.l Pens of any house in this country ; this 
being particularly true respecting quality and adaptation for 


SCHOOL PURPOSES. 


*** Parents , Teachers and Scholars who may wish to try these Pens , and 
who may not be able to obtain them through the trade, can receive a Card of 
Samples of the Spencerian by remitting to us 25 cents ; and the same amvunt 
for a box of Samples of the Perry Pens. 

IVISON, BLAKEMAN & COMPANY, 

New York and Chicago . 






















Approved Text-Books 

FOR 

HIGH SCHOOLS. 

-*♦«- 

T HERE are no text-books that require in their preparation so much prac¬ 
tical scholarship, combined with the teacher’s experience, as those com¬ 
piled for use in High Schools, Seminaries and Colleges. The treatment must 
be succinct yet thorough ; accuracy of statement, clearness of expression, and 
scientific gradation are indispensable. We have no special claims to make for 
our list on the score of the Ancient Classics, but in the modern languages, 
French, German and Spanish, in Botany, Geology, Chemistry and Astronomy, 
and the Higher Mathematics, Moral and Mental Science, etc., etc., we chal¬ 
lenge comparison with any competing books. Many of them are known to all 
scholars, and are reprinted abroad, while others have enjoyed a National repu¬ 
tation for many years. 


GUYOT'S WALL MAPS. 

GUYOT'S PHYSICAL GEOGRAPHY. 

WOODBURY'S GERMAN COURSE. Comprising a full series, from 
“ The Easy Lessons” to the most advanced manuals. 

FASQUELLE'S FRENCH COURSE. On the plan of Woodbury’s 
Method ; also a complete series. 

MIXER'S MANUAL OF FRENCH POETRY. 

LANGUELLIER & MONSANTO'S FRENCH GRAMMAR. 

HENNEQUIN'S FRENCH VERBS. 

MONSANTO'S FRENCH STUDENT'S ASSISTANT. 

MONSANTO & LANGUELLIER'S SPANISH GRAMMAR. 

GRAY'S BOTANICAL SERIES. 

DANA'S WORKS ON GEOLOGY. 

ELIOT & STORER'S CHEMISTRY. 

ROBINSON'S HIGHER MATHEMATICS. 

SWINTON'S OUTLINES OF HISTORY. 

FISHER'S OUTLINES OF HISTORY. 

CATHCART'S LITERARY READER. 

COOLEY'S WORKS ON NATURAL SCIENCE. 

TENNEY'S WORK ON ZOOLOGY AND NATURAL HISTORY. 
WELLS' WORKS ON NATURAL SCIENCE. 

KERL'S COMPREHENSIVE ENGLISH GRAMMAR. 

WEBSTER'S ACADEMIC DICTIONARY. 

TOWNSEND'S CIVIL GOVERNMENT. 

WHITE'S DRA WING. 

KUHNER'S GREEK GRAMMAR. 

KIDDLE'S ASTRONOMY. 

SWINTON'S COMPLETE GEOGRAPHY. Etc. . Etc. 


*** Descriptive Circulars and Price Lists will be sent to Teachers and Edu¬ 
cationists on application. Liberal terms will be made for introduction, exchange 
and examination. 


IVISON, BLAKEMAN & COMPANY, Publishers, 


B SPENCERIAN STEEL PENS 

ARE THE BEST. 


NEW YORK and CHICAGO. 


































